Integrand size = 24, antiderivative size = 733 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {\left (18 b^3 c d^2-144 a b c^2 d^2-8 b^4 e^2+57 a b^2 c e^2-84 a^2 c^2 e^2\right ) x \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b^2-4 a c\right ) d e \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{64 c^2}+\frac {x \left (9 b^2 c d^2+90 a c^2 d^2-4 b^3 e^2+9 a b c e^2+3 c \left (9 b c d^2-4 b^2 e^2+14 a c e^2\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {d e \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}+\frac {x \left (3 \left (3 c d^2+b e^2\right )+7 c e^2 x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac {3 \left (b^2-4 a c\right )^2 d e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{128 c^{5/2}}+\frac {\sqrt [4]{a} \left (18 b^3 c d^2-144 a b c^2 d^2-8 b^4 e^2+57 a b^2 c e^2-84 a^2 c^2 e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \left (18 b^2 c d^2-27 \sqrt {a} b c^{3/2} d^2-90 a c^2 d^2-8 b^3 e^2+12 \sqrt {a} b^2 \sqrt {c} e^2+33 a b c e^2-42 a^{3/2} c^{3/2} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a+b x^2+c x^4}} \] Output:
-1/315*(-84*a^2*c^2*e^2+57*a*b^2*c*e^2-144*a*b*c^2*d^2-8*b^4*e^2+18*b^3*c* d^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+c^(1/2)*x^2)-3/64*(-4*a*c+b^ 2)*d*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c^2+1/315*x*(9*b^2*c*d^2+90*a*c^2 *d^2-4*b^3*e^2+9*a*b*c*e^2+3*c*(14*a*c*e^2-4*b^2*e^2+9*b*c*d^2)*x^2)*(c*x^ 4+b*x^2+a)^(1/2)/c^2+1/8*d*e*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(3/2)/c+1/63*x*(7 *c*e^2*x^2+3*b*e^2+9*c*d^2)*(c*x^4+b*x^2+a)^(3/2)/c+3/128*(-4*a*c+b^2)^2*d *e*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(5/2)+1/315*a^ (1/4)*(-84*a^2*c^2*e^2+57*a*b^2*c*e^2-144*a*b*c^2*d^2-8*b^4*e^2+18*b^3*c*d ^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)* EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2) )/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)-1/630*a^(1/4)*(b+2*a^(1/2)*c^(1/2))*(18*b ^2*c*d^2-27*a^(1/2)*b*c^(3/2)*d^2-90*a*c^2*d^2-8*b^3*e^2+12*a^(1/2)*b^2*c^ (1/2)*e^2+33*a*b*c*e^2-42*a^(3/2)*c^(3/2)*e^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x ^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4 )*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(11/4)/(c*x^4+b*x^2+a)^(1/ 2)
Result contains complex when optimal does not.
Time = 15.10 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {32 i \left (-b+\sqrt {b^2-4 a c}\right ) \left (-18 b^3 c d^2+144 a b c^2 d^2+8 b^4 e^2-57 a b^2 c e^2+84 a^2 c^2 e^2\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-32 i \left (-8 b^5 e^2+12 a b c^2 \left (12 \sqrt {b^2-4 a c} d^2-11 a e^2\right )+b^3 c \left (-18 \sqrt {b^2-4 a c} d^2+65 a e^2\right )+2 b^4 \left (9 c d^2+4 \sqrt {b^2-4 a c} e^2\right )+12 a^2 c^2 \left (30 c d^2+7 \sqrt {b^2-4 a c} e^2\right )-3 a b^2 c \left (54 c d^2+19 \sqrt {b^2-4 a c} e^2\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+\sqrt {c} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-b^3 e (945 d+256 e x)+6 b^2 c x \left (96 d^2+105 d e x+32 e^2 x^2\right )+4 b c \left (3 a e (525 d+128 e x)+2 c x^3 \left (576 d^2+945 d e x+400 e^2 x^2\right )\right )+8 c^2 x \left (10 c x^4 \left (36 d^2+63 d e x+28 e^2 x^2\right )+a \left (1080 d^2+1575 d e x+616 e^2 x^2\right )\right )\right )-945 \left (b^2-4 a c\right )^2 d e \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{40320 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(d + e*x)^2*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
((32*I)*(-b + Sqrt[b^2 - 4*a*c])*(-18*b^3*c*d^2 + 144*a*b*c^2*d^2 + 8*b^4* e^2 - 57*a*b^2*c*e^2 + 84*a^2*c^2*e^2)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x ^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - (32*I)*(-8 *b^5*e^2 + 12*a*b*c^2*(12*Sqrt[b^2 - 4*a*c]*d^2 - 11*a*e^2) + b^3*c*(-18*S qrt[b^2 - 4*a*c]*d^2 + 65*a*e^2) + 2*b^4*(9*c*d^2 + 4*Sqrt[b^2 - 4*a*c]*e^ 2) + 12*a^2*c^2*(30*c*d^2 + 7*Sqrt[b^2 - 4*a*c]*e^2) - 3*a*b^2*c*(54*c*d^2 + 19*Sqrt[b^2 - 4*a*c]*e^2))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b ^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]* x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + Sqrt[c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(2*Sqrt[c]*(a + b*x^2 + c*x^4)*(-(b^3*e*(945*d + 256* e*x)) + 6*b^2*c*x*(96*d^2 + 105*d*e*x + 32*e^2*x^2) + 4*b*c*(3*a*e*(525*d + 128*e*x) + 2*c*x^3*(576*d^2 + 945*d*e*x + 400*e^2*x^2)) + 8*c^2*x*(10*c* x^4*(36*d^2 + 63*d*e*x + 28*e^2*x^2) + a*(1080*d^2 + 1575*d*e*x + 616*e^2* x^2))) - 945*(b^2 - 4*a*c)^2*d*e*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]]))/(40320*c^3*Sqrt[c/(b + Sqrt[b^2 - 4* a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 1.61 (sec) , antiderivative size = 702, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2202, 27, 1432, 1087, 1087, 1092, 219, 1490, 1490, 25, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\int 2 d e x \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+2 d e \int x \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+d e \int \left (c x^4+b x^2+a\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )+\int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )+\int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )+\int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \left (d^2+e^2 x^2\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\int \left (\left (9 b c d^2-4 b^2 e^2+14 a c e^2\right ) x^2+a \left (18 c d^2-b e^2\right )\right ) \sqrt {c x^4+b x^2+a}dx}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\frac {\int -\frac {\left (-8 e^2 b^4+18 c d^2 b^3+57 a c e^2 b^2-144 a c^2 d^2 b-84 a^2 c^2 e^2\right ) x^2+a \left (-4 e^2 b^3+9 c d^2 b^2+24 a c e^2 b-180 a c^2 d^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}-\frac {\int \frac {\left (-8 e^2 b^4+18 c d^2 b^3+57 a c e^2 b^2-144 a c^2 d^2 b-84 a^2 c^2 e^2\right ) x^2+a \left (-4 e^2 b^3+9 c d^2 b^2+24 a c e^2 b-180 a c^2 d^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (-42 a^{3/2} c^{3/2} e^2+12 \sqrt {a} b^2 \sqrt {c} e^2-27 \sqrt {a} b c^{3/2} d^2+33 a b c e^2-90 a c^2 d^2-8 b^3 e^2+18 b^2 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (-84 a^2 c^2 e^2+57 a b^2 c e^2-144 a b c^2 d^2-8 b^4 e^2+18 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (-42 a^{3/2} c^{3/2} e^2+12 \sqrt {a} b^2 \sqrt {c} e^2-27 \sqrt {a} b c^{3/2} d^2+33 a b c e^2-90 a c^2 d^2-8 b^3 e^2+18 b^2 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-144 a b c^2 d^2-8 b^4 e^2+18 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-42 a^{3/2} c^{3/2} e^2+12 \sqrt {a} b^2 \sqrt {c} e^2-27 \sqrt {a} b c^{3/2} d^2+33 a b c e^2-90 a c^2 d^2-8 b^3 e^2+18 b^2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-144 a b c^2 d^2-8 b^4 e^2+18 b^3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c e^2-4 b^2 e^2+9 b c d^2\right )+9 a b c e^2+90 a c^2 d^2-4 b^3 e^2+9 b^2 c d^2\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-42 a^{3/2} c^{3/2} e^2+12 \sqrt {a} b^2 \sqrt {c} e^2-27 \sqrt {a} b c^{3/2} d^2+33 a b c e^2-90 a c^2 d^2-8 b^3 e^2+18 b^2 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 e^2+57 a b^2 c e^2-144 a b c^2 d^2-8 b^4 e^2+18 b^3 c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+d e \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 \left (b e^2+3 c d^2\right )+7 c e^2 x^2\right )}{63 c}\) |
Input:
Int[(d + e*x)^2*(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(x*(3*(3*c*d^2 + b*e^2) + 7*c*e^2*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) + d*e*(((b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*( ((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(16*c)) + ( (x*(9*b^2*c*d^2 + 90*a*c^2*d^2 - 4*b^3*e^2 + 9*a*b*c*e^2 + 3*c*(9*b*c*d^2 - 4*b^2*e^2 + 14*a*c*e^2)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) - (-(((18*b ^3*c*d^2 - 144*a*b*c^2*d^2 - 8*b^4*e^2 + 57*a*b^2*c*e^2 - 84*a^2*c^2*e^2)* (-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a ] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip ticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4) *Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(1 8*b^2*c*d^2 - 27*Sqrt[a]*b*c^(3/2)*d^2 - 90*a*c^2*d^2 - 8*b^3*e^2 + 12*Sqr t[a]*b^2*Sqrt[c]*e^2 + 33*a*b*c*e^2 - 42*a^(3/2)*c^(3/2)*e^2)*(Sqrt[a] + S qrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[ 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sq rt[a + b*x^2 + c*x^4]))/(15*c))/(21*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 3.78 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.48
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1087\) |
| risch | \(\text {Expression too large to display}\) | \(1116\) |
| default | \(\text {Expression too large to display}\) | \(1269\) |
Input:
int((e*x+d)^2*(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/9*c*e^2*x^7*(c*x^4+b*x^2+a)^(1/2)+1/4*c*d*e*x^6*(c*x^4+b*x^2+a)^(1/2)+1/ 7*(10/9*b*c*e^2+c^2*d^2)/c*x^5*(c*x^4+b*x^2+a)^(1/2)+3/8*b*d*e*x^4*(c*x^4+ b*x^2+a)^(1/2)+1/5*(11/9*a*c*e^2+b^2*e^2+2*b*c*d^2-6/7*b/c*(10/9*b*c*e^2+c ^2*d^2))/c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/4*(5/2*a*c*d*e+1/8*b^2*d*e)/c*x^2*( c*x^4+b*x^2+a)^(1/2)+1/3*(2*a*b*e^2+2*a*c*d^2+b^2*d^2-4/5*b/c*(11/9*a*c*e^ 2+b^2*e^2+2*b*c*d^2-6/7*b/c*(10/9*b*c*e^2+c^2*d^2))-5/7*a/c*(10/9*b*c*e^2+ c^2*d^2))/c*x*(c*x^4+b*x^2+a)^(1/2)+1/2*(5/2*a*b*d*e-3/4*b/c*(5/2*a*c*d*e+ 1/8*b^2*d*e))/c*(c*x^4+b*x^2+a)^(1/2)+1/4*(a^2*d^2-1/3*a/c*(2*a*b*e^2+2*a* c*d^2+b^2*d^2-4/5*b/c*(11/9*a*c*e^2+b^2*e^2+2*b*c*d^2-6/7*b/c*(10/9*b*c*e^ 2+c^2*d^2))-5/7*a/c*(10/9*b*c*e^2+c^2*d^2)))*2^(1/2)/((-b+(-4*a*c+b^2)^(1/ 2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2 )^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+( -4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)) +1/2*(2*a^2*d*e-1/2*a/c*(5/2*a*c*d*e+1/8*b^2*d*e)-1/2*b/c*(5/2*a*b*d*e-3/4 *b/c*(5/2*a*c*d*e+1/8*b^2*d*e)))*ln((2*c*x^2+b)/c^(1/2)+2*(c*x^4+b*x^2+a)^ (1/2))/c^(1/2)-1/2*(a^2*e^2+2*a*b*d^2-3/5*a/c*(11/9*a*c*e^2+b^2*e^2+2*b*c* d^2-6/7*b/c*(10/9*b*c*e^2+c^2*d^2))-2/3*b/c*(2*a*b*e^2+2*a*c*d^2+b^2*d^2-4 /5*b/c*(11/9*a*c*e^2+b^2*e^2+2*b*c*d^2-6/7*b/c*(10/9*b*c*e^2+c^2*d^2))-5/7 *a/c*(10/9*b*c*e^2+c^2*d^2)))*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)* (4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a...
Time = 0.24 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/80640*(945*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(c)*d*e*x*log(8*c^2*x^ 4 + 8*b*c*x^2 + b^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4* a*c) - 128*sqrt(1/2)*((18*(b^3*c^2 - 8*a*b*c^3)*d^2 - (8*b^4*c - 57*a*b^2* c^2 + 84*a^2*c^3)*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (18*(b^4*c - 8*a*b^2*c^ 2)*d^2 - (8*b^5 - 57*a*b^3*c + 84*a^2*b*c^2)*e^2)*x)*sqrt(c)*sqrt((c*sqrt( (b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/( a*c)) + 128*sqrt(1/2)*((9*(2*b^3*c^2 + 20*a*c^4 - (16*a*b + b^2)*c^3)*d^2 - (8*b^4*c + 12*(7*a^2 + 2*a*b)*c^3 - (57*a*b^2 + 4*b^3)*c^2)*e^2)*x*sqrt( (b^2 - 4*a*c)/c^2) - (9*(2*b^4*c - 20*a*b*c^3 - (16*a*b^2 - b^3)*c^2)*d^2 - (8*b^5 + 12*(7*a^2*b - 2*a*b^2)*c^2 - (57*a*b^3 - 4*b^4)*c)*e^2)*x)*sqrt (c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sq rt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 4*(2240*c^5*e^2*x^8 + 5040*c^5*d*e*x^7 + 7560*b*c ^4*d*e*x^5 + 320*(9*c^5*d^2 + 10*b*c^4*e^2)*x^6 + 630*(b^2*c^3 + 20*a*c^4) *d*e*x^3 + 64*(72*b*c^4*d^2 + (3*b^2*c^3 + 77*a*c^4)*e^2)*x^4 - 315*(3*b^3 *c^2 - 20*a*b*c^3)*d*e*x - 1152*(b^3*c^2 - 8*a*b*c^3)*d^2 + 64*(8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3)*e^2 + 64*(9*(b^2*c^3 + 15*a*c^4)*d^2 - 4*(b^3*c ^2 - 6*a*b*c^3)*e^2)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^4*x)
\[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (d + e x\right )^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x+d)**2*(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral((d + e*x)**2*(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)^2, x)
\[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x + d)^2, x)
Timed out. \[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int((d + e*x)^2*(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int((d + e*x)^2*(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int (d+e x)^2 \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {too large to display} \] Input:
int((e*x+d)^2*(c*x^4+b*x^2+a)^(3/2),x)
Output:
(12600*sqrt(a + b*x**2 + c*x**4)*a*b*c**2*d*e + 3072*sqrt(a + b*x**2 + c*x **4)*a*b*c**2*e**2*x + 17280*sqrt(a + b*x**2 + c*x**4)*a*c**3*d**2*x + 252 00*sqrt(a + b*x**2 + c*x**4)*a*c**3*d*e*x**2 + 9856*sqrt(a + b*x**2 + c*x* *4)*a*c**3*e**2*x**3 - 1890*sqrt(a + b*x**2 + c*x**4)*b**3*c*d*e - 512*sqr t(a + b*x**2 + c*x**4)*b**3*c*e**2*x + 1152*sqrt(a + b*x**2 + c*x**4)*b**2 *c**2*d**2*x + 1260*sqrt(a + b*x**2 + c*x**4)*b**2*c**2*d*e*x**2 + 384*sqr t(a + b*x**2 + c*x**4)*b**2*c**2*e**2*x**3 + 9216*sqrt(a + b*x**2 + c*x**4 )*b*c**3*d**2*x**3 + 15120*sqrt(a + b*x**2 + c*x**4)*b*c**3*d*e*x**4 + 640 0*sqrt(a + b*x**2 + c*x**4)*b*c**3*e**2*x**5 + 5760*sqrt(a + b*x**2 + c*x* *4)*c**4*d**2*x**5 + 10080*sqrt(a + b*x**2 + c*x**4)*c**4*d*e*x**6 + 4480* sqrt(a + b*x**2 + c*x**4)*c**4*e**2*x**7 - 15120*sqrt(c)*log(sqrt(a + b*x* *2 + c*x**4) - sqrt(c)*x**2)*a**2*c**2*d*e + 7560*sqrt(c)*log(sqrt(a + b*x **2 + c*x**4) - sqrt(c)*x**2)*a*b**2*c*d*e - 945*sqrt(c)*log(sqrt(a + b*x* *2 + c*x**4) - sqrt(c)*x**2)*b**4*d*e + 15120*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a**2*c**2*d*e - 7560*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a*b**2*c*d*e + 945*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*b**4*d*e - 3072*int(sqrt(a + b*x**2 + c*x**4)/(a **2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**3*b*c**2*e**2 + 23040*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x **4 + b*c*x**6),x)*a**3*c**3*d**2 + 512*int(sqrt(a + b*x**2 + c*x**4)/(...