Integrand size = 30, antiderivative size = 459 \[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2 (189 A-306 B+26 C) \sqrt {2+6 x+3 x^2+9 x^3}}{8505}+\frac {2}{945} (189 A+9 B-44 C) x \sqrt {2+6 x+3 x^2+9 x^3}+\frac {4 (3213 A+468 B-818 C) \sqrt {2+6 x+3 x^2+9 x^3}}{8505 \left (1+\sqrt {7}+3 x\right )}+\frac {2}{189} (9 B-2 C) \left (2+3 x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3}+\frac {2}{81} C (1+3 x) \left (2+3 x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3}-\frac {4 (3213 A+468 B-818 C) \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \sqrt {2+6 x+3 x^2+9 x^3} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{1215 \sqrt {3} 7^{3/4} \sqrt {1+3 x} \left (2+3 x^2\right )}+\frac {2 \left (189 \left (17+\sqrt {7}\right ) A+468 B-306 \sqrt {7} B-818 C+26 \sqrt {7} C\right ) \left (1+\sqrt {7}+3 x\right ) \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \sqrt {2+6 x+3 x^2+9 x^3} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{1215 \sqrt {3} 7^{3/4} \sqrt {1+3 x} \left (2+3 x^2\right )} \] Output:
2/8505*(189*A-306*B+26*C)*(9*x^3+3*x^2+6*x+2)^(1/2)+2/945*(189*A+9*B-44*C) *x*(9*x^3+3*x^2+6*x+2)^(1/2)+4*(3213*A+468*B-818*C)*(9*x^3+3*x^2+6*x+2)^(1 /2)/(8505+8505*7^(1/2)+25515*x)+2/189*(9*B-2*C)*(3*x^2+2)*(9*x^3+3*x^2+6*x +2)^(1/2)+2/81*C*(1+3*x)*(3*x^2+2)*(9*x^3+3*x^2+6*x+2)^(1/2)-4/25515*(3213 *A+468*B-818*C)*(1+7^(1/2)+3*x)*((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3 +3*x^2+6*x+2)^(1/2)*EllipticE(sin(2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4))),1/1 4*(98+14*7^(1/2))^(1/2))*3^(1/2)*7^(1/4)/(1+3*x)^(1/2)/(3*x^2+2)+2/25515*( 189*(17+7^(1/2))*A+468*B-306*7^(1/2)*B-818*C+26*7^(1/2)*C)*(1+7^(1/2)+3*x) *((3*x^2+2)/(1+7^(1/2)+3*x)^2)^(1/2)*(9*x^3+3*x^2+6*x+2)^(1/2)*InverseJaco biAM(2*arctan(1/7*(1+3*x)^(1/2)*7^(3/4)),1/14*(98+14*7^(1/2))^(1/2))*3^(1/ 2)*7^(1/4)/(1+3*x)^(1/2)/(3*x^2+2)
Result contains complex when optimal does not.
Time = 7.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.59 \[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2 \left (2+3 x^2\right ) \left ((1+3 x) \sqrt {6+9 x^2} \left (189 A (1+9 x)+9 B \left (56+9 x+135 x^2\right )+C \left (56+234 x+45 x^2+945 x^3\right )\right )+\frac {2 i (3213 A+468 B-818 C) (1+3 x) E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )}{\sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}}}+14 i (189 A-306 B+26 C) \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )\right )}{8505 \sqrt {6+9 x^2} \sqrt {2+6 x+3 x^2+9 x^3}} \] Input:
Integrate[(A + B*x + C*x^2)*Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(2*(2 + 3*x^2)*((1 + 3*x)*Sqrt[6 + 9*x^2]*(189*A*(1 + 9*x) + 9*B*(56 + 9*x + 135*x^2) + C*(56 + 234*x + 45*x^2 + 945*x^3)) + ((2*I)*(3213*A + 468*B - 818*C)*(1 + 3*x)*EllipticE[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/ 4))], (2*Sqrt[6])/(I + Sqrt[6])])/Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])] + (14* I)*(189*A - 306*B + 26*C)*Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])]*EllipticF[ArcS in[Sqrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])]) )/(8505*Sqrt[6 + 9*x^2]*Sqrt[2 + 6*x + 3*x^2 + 9*x^3])
Result contains complex when optimal does not.
Time = 4.24 (sec) , antiderivative size = 1282, normalized size of antiderivative = 2.79, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2526, 27, 2490, 2486, 27, 1236, 27, 1231, 27, 1269, 1172, 321, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {9 x^3+3 x^2+6 x+2} \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {1}{27} \int 3 (9 A-2 C+(9 B-2 C) x) \sqrt {9 x^3+3 x^2+6 x+2}dx+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int (9 A-2 C+(9 B-2 C) x) \sqrt {9 x^3+3 x^2+6 x+2}dx+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {1}{9} \int \left (\frac {1}{27} (27 (9 A-2 C)-3 (9 B-2 C))+(9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \sqrt {9 \left (x+\frac {1}{9}\right )^3+\frac {17}{3} \left (x+\frac {1}{9}\right )+\frac {110}{81}}d\left (x+\frac {1}{9}\right )+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 2486 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \int \frac {1}{9} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}d\left (x+\frac {1}{9}\right )}{81 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \int \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 A-9 B-16 C+9 (9 B-2 C) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}d\left (x+\frac {1}{9}\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {2}{567} \int \frac {81 \left (\frac {567 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (193+441 \sqrt {2}-\left (229-126 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-119 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (3658-3528 \sqrt {2}-119 \left (-55+63 \sqrt {2}\right )^{2/3}+2 \sqrt [3]{-55+63 \sqrt {2}} \left (421+63 \sqrt {2}\right )\right ) C}{\left (-55+63 \sqrt {2}\right )^{2/3}}+9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{2 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}d\left (x+\frac {1}{9}\right )+\frac {2}{63} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} (9 B-2 C)\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {1}{7} \int \frac {\left (\frac {567 \left (55-63 \sqrt {2}+17 \sqrt [3]{-55+63 \sqrt {2}}\right ) A+9 \left (193+441 \sqrt {2}-\left (229-126 \sqrt {2}\right ) \sqrt [3]{-55+63 \sqrt {2}}-119 \left (-55+63 \sqrt {2}\right )^{2/3}\right ) B-2 \left (3658-3528 \sqrt {2}-119 \left (-55+63 \sqrt {2}\right )^{2/3}+2 \sqrt [3]{-55+63 \sqrt {2}} \left (421+63 \sqrt {2}\right )\right ) C}{\left (-55+63 \sqrt {2}\right )^{2/3}}+9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}d\left (x+\frac {1}{9}\right )+\frac {2}{63} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} (9 B-2 C)\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {1}{7} \left (-\frac {2 \int -\frac {177147 \left (5 (2079 A-1098 B-218 C)+9 (3213 A+468 B-818 C) \left (x+\frac {1}{9}\right )\right )}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )}{98415}-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17} \left (\frac {5 \left (289-17 \left (63 \sqrt {2}-55\right )^{2/3}+\left (63 \sqrt {2}-55\right )^{4/3}\right ) (9 B-2 C)}{\left (63 \sqrt {2}-55\right )^{2/3}}-9 \left (x+\frac {1}{9}\right ) \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{63 \sqrt {2}-55}}+5 \sqrt [3]{63 \sqrt {2}-55}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{63 \sqrt {2}-55}}-5 \sqrt [3]{63 \sqrt {2}-55}\right ) C\right )\right )\right )+\frac {2}{63} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} (9 B-2 C)\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {1}{7} \left (\frac {18}{5} \int \frac {5 (2079 A-1098 B-218 C)+9 (3213 A+468 B-818 C) \left (x+\frac {1}{9}\right )}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17} \left (\frac {5 \left (289-17 \left (63 \sqrt {2}-55\right )^{2/3}+\left (63 \sqrt {2}-55\right )^{4/3}\right ) (9 B-2 C)}{\left (63 \sqrt {2}-55\right )^{2/3}}-9 \left (x+\frac {1}{9}\right ) \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{63 \sqrt {2}-55}}+5 \sqrt [3]{63 \sqrt {2}-55}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{63 \sqrt {2}-55}}-5 \sqrt [3]{63 \sqrt {2}-55}\right ) C\right )\right )\right )+\frac {2}{63} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17\right )^{3/2} (9 B-2 C)\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (63 \sqrt {2}-55\right )^{2/3}}{\sqrt [3]{63 \sqrt {2}-55}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (63 \sqrt {2}-55\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{63 \sqrt {2}-55}}+\left (63 \sqrt {2}-55\right )^{2/3}+\frac {289}{\left (63 \sqrt {2}-55\right )^{2/3}}+17}}+\frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}+\frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {2}{63} (9 B-2 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}+\frac {1}{7} \left (\frac {18}{5} \left ((3213 A+468 B-818 C) \int \frac {\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}{\sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )-\left (\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (3213 A+468 B-818 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-5 (2079 A-1098 B-218 C)\right ) \int \frac {1}{\sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}d\left (x+\frac {1}{9}\right )\right )-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {5 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) (9 B-2 C)}{\left (-55+63 \sqrt {2}\right )^{2/3}}-9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}\right )\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}+\frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {2}{63} (9 B-2 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}+\frac {1}{7} \left (\frac {18}{5} \left (\frac {i \sqrt {2} (3213 A+468 B-818 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}-\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (3213 A+468 B-818 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-5 (2079 A-1098 B-218 C)\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \int \frac {1}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1} \sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {5 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) (9 B-2 C)}{\left (-55+63 \sqrt {2}\right )^{2/3}}-9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}\right )\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}+\frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {2}{63} (9 B-2 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}+\frac {1}{7} \left (\frac {18}{5} \left (\frac {i \sqrt {2} (3213 A+468 B-818 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \int \frac {\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\sqrt {3} \left (17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )\right )}+1}}{\sqrt {\frac {i \sqrt [3]{-55+63 \sqrt {2}} \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{2 \sqrt {3} \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}+1}}d\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}}{9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}-\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (3213 A+468 B-818 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-5 (2079 A-1098 B-218 C)\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {5 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) (9 B-2 C)}{\left (-55+63 \sqrt {2}\right )^{2/3}}-9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}\right )\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{81} C \left (9 x^3+3 x^2+6 x+2\right )^{3/2}+\frac {\sqrt {729 \left (x+\frac {1}{9}\right )^3+459 \left (x+\frac {1}{9}\right )+110} \left (\frac {2}{63} (9 B-2 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17\right )^{3/2}+\frac {1}{7} \left (\frac {18}{5} \left (\frac {i \sqrt {2} (3213 A+468 B-818 C) \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} E\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right )|\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \sqrt [6]{-55+63 \sqrt {2}} \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}}}-\frac {2 i \sqrt {2} \sqrt [6]{-55+63 \sqrt {2}} \left (\frac {\left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) (3213 A+468 B-818 C)}{\sqrt [3]{-55+63 \sqrt {2}}}-5 (2079 A-1098 B-218 C)\right ) \sqrt {-\frac {i \left (9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}{\left (-55+63 \sqrt {2}\right )^{2/3} \left (3 i-\sqrt {3}\right )-17 \left (3 i+\sqrt {3}\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [6]{-55+63 \sqrt {2}} \sqrt {-i \left (18 \left (x+\frac {1}{9}\right )+\frac {\left (-55+63 \sqrt {2}\right )^{2/3} \left (1+i \sqrt {3}\right )+17 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}\right )}}{\sqrt [4]{3} \sqrt {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}}\right ),\frac {2 \left (17+\left (-55+63 \sqrt {2}\right )^{2/3}\right )}{17+17 i \sqrt {3}+\left (-55+63 \sqrt {2}\right )^{2/3} \left (1-i \sqrt {3}\right )}\right )}{9 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}}}\right )-\frac {2}{45} \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \left (\frac {5 \left (289-17 \left (-55+63 \sqrt {2}\right )^{2/3}+\left (-55+63 \sqrt {2}\right )^{4/3}\right ) (9 B-2 C)}{\left (-55+63 \sqrt {2}\right )^{2/3}}-9 \left (567 A-9 \left (7-\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}+5 \sqrt [3]{-55+63 \sqrt {2}}\right ) B-2 \left (56+\frac {85}{\sqrt [3]{-55+63 \sqrt {2}}}-5 \sqrt [3]{-55+63 \sqrt {2}}\right ) C\right ) \left (x+\frac {1}{9}\right )\right ) \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}\right )\right )}{729 \sqrt {9 \left (x+\frac {1}{9}\right )+\frac {17-\left (-55+63 \sqrt {2}\right )^{2/3}}{\sqrt [3]{-55+63 \sqrt {2}}}} \sqrt {81 \left (x+\frac {1}{9}\right )^2-\frac {9 \left (17-\left (-55+63 \sqrt {2}\right )^{2/3}\right ) \left (x+\frac {1}{9}\right )}{\sqrt [3]{-55+63 \sqrt {2}}}+\left (-55+63 \sqrt {2}\right )^{2/3}+\frac {289}{\left (-55+63 \sqrt {2}\right )^{2/3}}+17}}\) |
Input:
Int[(A + B*x + C*x^2)*Sqrt[2 + 6*x + 3*x^2 + 9*x^3],x]
Output:
(2*C*(2 + 6*x + 3*x^2 + 9*x^3)^(3/2))/81 + (Sqrt[110 + 459*(1/9 + x) + 729 *(1/9 + x)^3]*((2*(9*B - 2*C)*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)]*(17 + 289/(-55 + 63*Sqrt[2])^(2/3) + (-55 + 63*Sqrt[2])^(2/3) - (9*(17 - (-55 + 63*Sqrt[2])^(2/3))*(1/9 + x))/(-55 + 63*Sqrt[2])^(1/3) + 81*(1/9 + x)^2)^(3/2))/63 + ((-2*Sqrt[(17 - (-55 + 6 3*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x)]*((5*(289 - 17*(- 55 + 63*Sqrt[2])^(2/3) + (-55 + 63*Sqrt[2])^(4/3))*(9*B - 2*C))/(-55 + 63* Sqrt[2])^(2/3) - 9*(567*A - 9*(7 - 85/(-55 + 63*Sqrt[2])^(1/3) + 5*(-55 + 63*Sqrt[2])^(1/3))*B - 2*(56 + 85/(-55 + 63*Sqrt[2])^(1/3) - 5*(-55 + 63*S qrt[2])^(1/3))*C)*(1/9 + x))*Sqrt[17 + 289/(-55 + 63*Sqrt[2])^(2/3) + (-55 + 63*Sqrt[2])^(2/3) - (9*(17 - (-55 + 63*Sqrt[2])^(2/3))*(1/9 + x))/(-55 + 63*Sqrt[2])^(1/3) + 81*(1/9 + x)^2])/45 + (18*(((I/9)*Sqrt[2]*(3213*A + 468*B - 818*C)*Sqrt[(17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/ 3) + 9*(1/9 + x)]*EllipticE[ArcSin[((-55 + 63*Sqrt[2])^(1/6)*Sqrt[(-I)*((( -55 + 63*Sqrt[2])^(2/3)*(1 + I*Sqrt[3]) + (17*I)*(I + Sqrt[3]))/(-55 + 63* Sqrt[2])^(1/3) + 18*(1/9 + x))])/(3^(1/4)*Sqrt[2*(17 + (-55 + 63*Sqrt[2])^ (2/3))])], (2*(17 + (-55 + 63*Sqrt[2])^(2/3)))/(17 + (17*I)*Sqrt[3] + (-55 + 63*Sqrt[2])^(2/3)*(1 - I*Sqrt[3]))])/((-55 + 63*Sqrt[2])^(1/6)*Sqrt[((- I)*((17 - (-55 + 63*Sqrt[2])^(2/3))/(-55 + 63*Sqrt[2])^(1/3) + 9*(1/9 + x) ))/((-55 + 63*Sqrt[2])^(2/3)*(3*I - Sqrt[3]) - 17*(3*I + Sqrt[3]))]) - ...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} , Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x ]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p) Int[(e + f*x)^m*Sim p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 *(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* b^3 + 27*a^2*d, 0] && !IntegerQ[p]
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] }, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp [1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x , m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm , x] && PolyQ[Qn, x] && NeQ[p, -1]
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.94
| method | result | size |
| elliptic | \(\frac {2 C \,x^{3} \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9}+\left (\frac {2 B}{7}+\frac {2 C}{189}\right ) x^{2} \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {2 B}{105}+\frac {2 A}{5}+\frac {52 C}{945}\right ) x \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\left (\frac {16 B}{135}+\frac {2 A}{45}+\frac {16 C}{1215}\right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}+\frac {2 \left (\frac {16 A}{15}-\frac {424 C}{2835}-\frac {124 B}{315}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {2 \left (\frac {104 B}{315}+\frac {34 A}{15}-\frac {1636 C}{2835}\right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(430\) |
| risch | \(\frac {2 \left (945 C \,x^{3}+1215 B \,x^{2}+45 C \,x^{2}+1701 A x +81 B x +234 C x +189 A +504 B +56 C \right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{8505}+\frac {4 \left (3213 A +468 B -818 C \right ) \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{2835 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {32 A \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{15 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {248 B \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{315 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}-\frac {848 C \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{2835 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(620\) |
| default | \(\text {Expression too large to display}\) | \(1094\) |
Input:
int((C*x^2+B*x+A)*(9*x^3+3*x^2+6*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/9*C*x^3*(9*x^3+3*x^2+6*x+2)^(1/2)+(2/7*B+2/189*C)*x^2*(9*x^3+3*x^2+6*x+2 )^(1/2)+(2/105*B+2/5*A+52/945*C)*x*(9*x^3+3*x^2+6*x+2)^(1/2)+(16/135*B+2/4 5*A+16/1215*C)*(9*x^3+3*x^2+6*x+2)^(1/2)+2*(16/15*A-424/2835*C-124/315*B)* (-1/3*I*6^(1/2)+1/3)*((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2)*((x-1/3*I*6^(1/2 ))/(-1/3-1/3*I*6^(1/2)))^(1/2)*((x+1/3*I*6^(1/2))/(-1/3+1/3*I*6^(1/2)))^(1 /2)/(9*x^3+3*x^2+6*x+2)^(1/2)*EllipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/ 2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2))+2*(104/315*B+34/15*A -1636/2835*C)*(-1/3*I*6^(1/2)+1/3)*((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2)*(( x-1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)*((x+1/3*I*6^(1/2))/(-1/3+1/3* I*6^(1/2)))^(1/2)/(9*x^3+3*x^2+6*x+2)^(1/2)*((-1/3-1/3*I*6^(1/2))*Elliptic E(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6 ^(1/2)))^(1/2))+1/3*I*6^(1/2)*EllipticF(((x+1/3)/(-1/3*I*6^(1/2)+1/3))^(1/ 2),((-1/3+1/3*I*6^(1/2))/(-1/3-1/3*I*6^(1/2)))^(1/2)))
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.21 \[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {4}{15309} \, {\left (2079 \, A - 1098 \, B - 218 \, C\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) - \frac {4}{8505} \, {\left (3213 \, A + 468 \, B - 818 \, C\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) + \frac {2}{8505} \, {\left (945 \, C x^{3} + 45 \, {\left (27 \, B + C\right )} x^{2} + 9 \, {\left (189 \, A + 9 \, B + 26 \, C\right )} x + 189 \, A + 504 \, B + 56 \, C\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \] Input:
integrate((C*x^2+B*x+A)*(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="fricas")
Output:
4/15309*(2079*A - 1098*B - 218*C)*weierstrassPInverse(-68/27, -440/729, x + 1/9) - 4/8505*(3213*A + 468*B - 818*C)*weierstrassZeta(-68/27, -440/729, weierstrassPInverse(-68/27, -440/729, x + 1/9)) + 2/8505*(945*C*x^3 + 45* (27*B + C)*x^2 + 9*(189*A + 9*B + 26*C)*x + 189*A + 504*B + 56*C)*sqrt(9*x ^3 + 3*x^2 + 6*x + 2)
\[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int \sqrt {\left (3 x + 1\right ) \left (3 x^{2} + 2\right )} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(9*x**3+3*x**2+6*x+2)**(1/2),x)
Output:
Integral(sqrt((3*x + 1)*(3*x**2 + 2))*(A + B*x + C*x**2), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int { {\left (C x^{2} + B x + A\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\int { {\left (C x^{2} + B x + A\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(9*x^3+3*x^2+6*x+2)^(1/2),x, algorithm="giac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(9*x^3 + 3*x^2 + 6*x + 2), x)
Time = 12.45 (sec) , antiderivative size = 1846, normalized size of antiderivative = 4.02 \[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\text {Too large to display} \] Input:
int((A + B*x + C*x^2)*(6*x + 3*x^2 + 9*x^3 + 2)^(1/2),x)
Output:
(((9*B)/7 + (3*C)/7)*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2)*(2*x^2*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2) - (20*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2))/9 - ( (4*x)/5 - 16/45)*((2*x)/3 + x^2/3 + x^3 + 2/9)^(1/2) + (16*((6^(1/2)*1i)/3 + 1/3)*(((6^(1/2)*1i)/3 - 1/3)*ellipticE(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3)) - (6^(1/2) *ellipticF(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/ 3 + 1/3)/((6^(1/2)*1i)/3 - 1/3))*1i)/3)*((x + (6^(1/2)*1i)/3)/((6^(1/2)*1i )/3 - 1/3))^(1/2)*(-(x - (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)*((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2))/(15*((2*x)/3 + x^2/3 + x^3 + 2/9)^( 1/2)) + (8*((6^(1/2)*1i)/3 + 1/3)*((x + (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 - 1/3))^(1/2)*(-(x - (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)*((x + 1/3 )/((6^(1/2)*1i)/3 + 1/3))^(1/2)*ellipticF(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3)))/(5*((2*x) /3 + x^2/3 + x^3 + 2/9)^(1/2))))/(6*x + 3*x^2 + 9*x^3 + 2)^(1/2) - (((4*(( 6^(1/2)*1i)/3 + 1/3)*(((6^(1/2)*1i)/3 - 1/3)*ellipticE(asin(((x + 1/3)/((6 ^(1/2)*1i)/3 + 1/3))^(1/2)), -((6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3 )) - (6^(1/2)*ellipticF(asin(((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2)), -( (6^(1/2)*1i)/3 + 1/3)/((6^(1/2)*1i)/3 - 1/3))*1i)/3)*((x + (6^(1/2)*1i)/3) /((6^(1/2)*1i)/3 - 1/3))^(1/2)*(-(x - (6^(1/2)*1i)/3)/((6^(1/2)*1i)/3 + 1/ 3))^(1/2)*((x + 1/3)/((6^(1/2)*1i)/3 + 1/3))^(1/2))/(9*((2*x)/3 + x^2/3...
\[ \int \left (A+B x+C x^2\right ) \sqrt {2+6 x+3 x^2+9 x^3} \, dx=\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a x}{5}+\frac {4 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, a}{5}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b \,x^{2}}{7}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b x}{105}+\frac {8 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, b}{35}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{3}}{9}+\frac {2 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c \,x^{2}}{189}+\frac {52 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c x}{945}-\frac {508 \sqrt {9 x^{3}+3 x^{2}+6 x +2}\, c}{2835}-\frac {6 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) a}{5}-\frac {76 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b}{105}+\frac {404 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) c}{945}-\frac {51 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) a}{5}-\frac {52 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) b}{35}+\frac {818 \left (\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}\, x^{2}}{9 x^{3}+3 x^{2}+6 x +2}d x \right ) c}{315} \] Input:
int((C*x^2+B*x+A)*(9*x^3+3*x^2+6*x+2)^(1/2),x)
Output:
(1134*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*a*x + 2268*sqrt(9*x**3 + 3*x**2 + 6* x + 2)*a + 810*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b*x**2 + 54*sqrt(9*x**3 + 3 *x**2 + 6*x + 2)*b*x + 648*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*b + 630*sqrt(9* x**3 + 3*x**2 + 6*x + 2)*c*x**3 + 30*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x** 2 + 156*sqrt(9*x**3 + 3*x**2 + 6*x + 2)*c*x - 508*sqrt(9*x**3 + 3*x**2 + 6 *x + 2)*c - 3402*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(9*x**3 + 3*x**2 + 6* x + 2),x)*a - 2052*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*b + 1212*int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*c - 28917*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*a - 4212*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2 )/(9*x**3 + 3*x**2 + 6*x + 2),x)*b + 7362*int((sqrt(9*x**3 + 3*x**2 + 6*x + 2)*x**2)/(9*x**3 + 3*x**2 + 6*x + 2),x)*c)/2835