Integrand size = 19, antiderivative size = 447 \[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\frac {(2+x) (1+3 x) \left (2+3 x^2\right )}{42 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {(1+3 x) (51+50 x) \left (2+3 x^2\right )^2}{588 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {205 (1+3 x) \left (2+3 x^2\right )^3}{4116 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {935 (1+3 x)^2 \left (2+3 x^2\right )^3}{14406 \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {935 (1+3 x)^3 \left (2+3 x^2\right )^3}{14406 \left (1+\sqrt {7}+3 x\right ) \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}-\frac {935 (1+3 x)^{5/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right )^2 \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2058 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{5/2}}+\frac {5 \left (374-41 \sqrt {7}\right ) (1+3 x)^{5/2} \left (1+\sqrt {7}+3 x\right ) \left (2+3 x^2\right )^2 \sqrt {\frac {2+3 x^2}{\left (1+\sqrt {7}+3 x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {1+3 x}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{8232 \sqrt {3} 7^{3/4} \left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \] Output:
1/42*(2+x)*(1+3*x)*(3*x^2+2)/(9*x^3+3*x^2+6*x+2)^(5/2)+1/588*(1+3*x)*(51+5 0*x)*(3*x^2+2)^2/(9*x^3+3*x^2+6*x+2)^(5/2)-205/4116*(1+3*x)*(3*x^2+2)^3/(9 *x^3+3*x^2+6*x+2)^(5/2)-935/14406*(1+3*x)^2*(3*x^2+2)^3/(9*x^3+3*x^2+6*x+2 )^(5/2)+935/14406*(1+3*x)^3*(3*x^2+2)^3/(1+7^(1/2)+3*x)/(9*x^3+3*x^2+6*x+2 )^(5/2)-935/43218*(1+3*x)^(5/2)*(1+7^(1/2)+3*x)*(3*x^2+2)^2*((3*x^2+2)/(1+ 7^(1/2)+3*x)^2)^(1/2)*EllipticE(sin(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)/(9*x^3+3*x^2+6*x+2)^(5/2)+5/172 872*(374-41*7^(1/2))*(1+3*x)^(5/2)*(1+7^(1/2)+3*x)*(3*x^2+2)^2*((3*x^2+2)/ (1+7^(1/2)+3*x)^2)^(1/2)*InverseJacobiAM(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)/(9*x^3+3*x^2+6*x+2)^(5/2)
Result contains complex when optimal does not.
Time = 9.47 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\frac {-98 \left (20-7 x+51 x^2\right )-5 \left (489+287 x+1122 x^2\right ) \left (2+6 x+3 x^2+9 x^3\right )+\frac {5}{3} \sqrt {\frac {i (1+3 x)}{i+\sqrt {6}}} \sqrt {6+9 x^2} \left (2+6 x+3 x^2+9 x^3\right ) \left (374 \left (i+\sqrt {6}\right ) E\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right )|\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )-287 i \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {6}-3 i x}}{2^{3/4} \sqrt [4]{3}}\right ),\frac {2 \sqrt {6}}{i+\sqrt {6}}\right )\right )}{28812 \left (2+6 x+3 x^2+9 x^3\right )^{3/2}} \] Input:
Integrate[(2 + 6*x + 3*x^2 + 9*x^3)^(-5/2),x]
Output:
(-98*(20 - 7*x + 51*x^2) - 5*(489 + 287*x + 1122*x^2)*(2 + 6*x + 3*x^2 + 9 *x^3) + (5*Sqrt[(I*(1 + 3*x))/(I + Sqrt[6])]*Sqrt[6 + 9*x^2]*(2 + 6*x + 3* x^2 + 9*x^3)*(374*(I + Sqrt[6])*EllipticE[ArcSin[Sqrt[Sqrt[6] - (3*I)*x]/( 2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])] - (287*I)*EllipticF[ArcSin[S qrt[Sqrt[6] - (3*I)*x]/(2^(3/4)*3^(1/4))], (2*Sqrt[6])/(I + Sqrt[6])]))/3) /(28812*(2 + 6*x + 3*x^2 + 9*x^3)^(3/2))
Time = 0.55 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.95, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {2477, 27, 496, 27, 686, 27, 688, 27, 688, 27, 599, 27, 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.
| \(\displaystyle \int \frac {1}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2477 |
| \(\displaystyle \frac {243 (3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \int \frac {1}{243 (3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2}}dx}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \int \frac {1}{(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2}}dx}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}-\frac {1}{126} \int -\frac {3 (21 x+58)}{2 (3 x+1)^{5/2} \left (3 x^2+2\right )^{3/2}}dx\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \int \frac {21 x+58}{(3 x+1)^{5/2} \left (3 x^2+2\right )^{3/2}}dx+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}-\frac {1}{126} \int -\frac {405 (10 x+17)}{(3 x+1)^{5/2} \sqrt {3 x^2+2}}dx\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \int \frac {10 x+17}{(3 x+1)^{5/2} \sqrt {3 x^2+2}}dx+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (-\frac {2}{63} \int -\frac {3 (111-41 x)}{2 (3 x+1)^{3/2} \sqrt {3 x^2+2}}dx-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \int \frac {111-41 x}{(3 x+1)^{3/2} \sqrt {3 x^2+2}}dx-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2}{21} \int -\frac {3 (374 x+29)}{2 \sqrt {3 x+1} \sqrt {3 x^2+2}}dx-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (\frac {1}{7} \int \frac {374 x+29}{\sqrt {3 x+1} \sqrt {3 x^2+2}}dx-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2}{63} \int \frac {\sqrt {3} (287-374 (3 x+1))}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2 \int \frac {287-374 (3 x+1)}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}}{21 \sqrt {3}}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2 \left (\left (287-374 \sqrt {7}\right ) \int \frac {1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}+374 \sqrt {7} \int \frac {-3 x+\sqrt {7}-1}{\sqrt {7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}\right )}{21 \sqrt {3}}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2 \left (\left (287-374 \sqrt {7}\right ) \int \frac {1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}+374 \int \frac {-3 x+\sqrt {7}-1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}\right )}{21 \sqrt {3}}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2 \left (374 \int \frac {-3 x+\sqrt {7}-1}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}d\sqrt {3 x+1}+\frac {\left (287-374 \sqrt {7}\right ) \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}\right )}{21 \sqrt {3}}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {(3 x+1)^{5/2} \left (3 x^2+2\right )^{5/2} \left (\frac {1}{84} \left (\frac {45}{14} \left (\frac {1}{21} \left (-\frac {2 \left (\frac {\left (287-374 \sqrt {7}\right ) \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}+374 \left (\frac {\sqrt [4]{7} \left (3 x+\sqrt {7}+1\right ) \sqrt {\frac {(3 x+1)^2-2 (3 x+1)+7}{\left (3 x+\sqrt {7}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {3 x+1}}{\sqrt [4]{7}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{\sqrt {(3 x+1)^2-2 (3 x+1)+7}}-\frac {\sqrt {3 x+1} \sqrt {(3 x+1)^2-2 (3 x+1)+7}}{3 x+\sqrt {7}+1}\right )\right )}{21 \sqrt {3}}-\frac {748 \sqrt {3 x^2+2}}{21 \sqrt {3 x+1}}\right )-\frac {82 \sqrt {3 x^2+2}}{63 (3 x+1)^{3/2}}\right )+\frac {50 x+51}{7 (3 x+1)^{3/2} \sqrt {3 x^2+2}}\right )+\frac {x+2}{42 (3 x+1)^{3/2} \left (3 x^2+2\right )^{3/2}}\right )}{\left (9 x^3+3 x^2+6 x+2\right )^{5/2}}\) |
Input:
Int[(2 + 6*x + 3*x^2 + 9*x^3)^(-5/2),x]
Output:
((1 + 3*x)^(5/2)*(2 + 3*x^2)^(5/2)*((2 + x)/(42*(1 + 3*x)^(3/2)*(2 + 3*x^2 )^(3/2)) + ((51 + 50*x)/(7*(1 + 3*x)^(3/2)*Sqrt[2 + 3*x^2]) + (45*((-82*Sq rt[2 + 3*x^2])/(63*(1 + 3*x)^(3/2)) + ((-748*Sqrt[2 + 3*x^2])/(21*Sqrt[1 + 3*x]) - (2*(374*(-((Sqrt[1 + 3*x]*Sqrt[7 - 2*(1 + 3*x) + (1 + 3*x)^2])/(1 + Sqrt[7] + 3*x)) + (7^(1/4)*(1 + Sqrt[7] + 3*x)*Sqrt[(7 - 2*(1 + 3*x) + (1 + 3*x)^2)/(1 + Sqrt[7] + 3*x)^2]*EllipticE[2*ArcTan[Sqrt[1 + 3*x]/7^(1/ 4)], (7 + Sqrt[7])/14])/Sqrt[7 - 2*(1 + 3*x) + (1 + 3*x)^2]) + ((287 - 374 *Sqrt[7])*(1 + Sqrt[7] + 3*x)*Sqrt[(7 - 2*(1 + 3*x) + (1 + 3*x)^2)/(1 + Sq rt[7] + 3*x)^2]*EllipticF[2*ArcTan[Sqrt[1 + 3*x]/7^(1/4)], (7 + Sqrt[7])/1 4])/(2*7^(1/4)*Sqrt[7 - 2*(1 + 3*x) + (1 + 3*x)^2])))/(21*Sqrt[3]))/21))/1 4)/84))/(2 + 6*x + 3*x^2 + 9*x^3)^(5/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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[((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[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c + d*x)^p*(b + d* x^2)^p) Int[(c + d*x)^p*(b + d*x^2)^p, x], x] /; EqQ[b*c - a*d, 0]] /; Fr eeQ[p, x] && PolyQ[Px, x, 3] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {50490 x^{5}+29745 x^{4}+59970 x^{3}+32163 x^{2}+16854 x +6850}{28812 \left (9 x^{3}+3 x^{2}+6 x +2\right )^{\frac {3}{2}}}+\frac {145 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{9604 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {935 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{4802 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(346\) |
| default | \(\frac {\left (-\frac {10}{11907}+\frac {1}{3402} x -\frac {17}{7938} x^{2}\right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{\left (x^{3}+\frac {1}{3} x^{2}+\frac {2}{3} x +\frac {2}{9}\right )^{2}}-\frac {18 \left (\frac {815}{172872}+\frac {935}{86436} x^{2}+\frac {205}{74088} x \right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {145 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{9604 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {935 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{4802 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(374\) |
| elliptic | \(\frac {\left (-\frac {10}{11907}+\frac {1}{3402} x -\frac {17}{7938} x^{2}\right ) \sqrt {9 x^{3}+3 x^{2}+6 x +2}}{\left (x^{3}+\frac {1}{3} x^{2}+\frac {2}{3} x +\frac {2}{9}\right )^{2}}-\frac {18 \left (\frac {815}{172872}+\frac {935}{86436} x^{2}+\frac {205}{74088} x \right )}{\sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {145 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{9604 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}+\frac {935 \left (-\frac {i \sqrt {6}}{3}+\frac {1}{3}\right ) \sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}\, \sqrt {\frac {x -\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\, \sqrt {\frac {x +\frac {i \sqrt {6}}{3}}{-\frac {1}{3}+\frac {i \sqrt {6}}{3}}}\, \left (\left (-\frac {1}{3}-\frac {i \sqrt {6}}{3}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )+\frac {i \sqrt {6}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {1}{3}}{-\frac {i \sqrt {6}}{3}+\frac {1}{3}}}, \sqrt {\frac {-\frac {1}{3}+\frac {i \sqrt {6}}{3}}{-\frac {1}{3}-\frac {i \sqrt {6}}{3}}}\right )}{3}\right )}{4802 \sqrt {9 x^{3}+3 x^{2}+6 x +2}}\) | \(374\) |
Input:
int(1/(9*x^3+3*x^2+6*x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/28812*(50490*x^5+29745*x^4+59970*x^3+32163*x^2+16854*x+6850)/(9*x^3+3*x ^2+6*x+2)^(3/2)+145/9604*(-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))+935/4802*(-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 = 0.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=-\frac {565 \, {\left (81 \, x^{6} + 54 \, x^{5} + 117 \, x^{4} + 72 \, x^{3} + 48 \, x^{2} + 24 \, x + 4\right )} {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right ) + 16830 \, {\left (81 \, x^{6} + 54 \, x^{5} + 117 \, x^{4} + 72 \, x^{3} + 48 \, x^{2} + 24 \, x + 4\right )} {\rm weierstrassZeta}\left (-\frac {68}{27}, -\frac {440}{729}, {\rm weierstrassPInverse}\left (-\frac {68}{27}, -\frac {440}{729}, x + \frac {1}{9}\right )\right ) + 9 \, {\left (50490 \, x^{5} + 29745 \, x^{4} + 59970 \, x^{3} + 32163 \, x^{2} + 16854 \, x + 6850\right )} \sqrt {9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2}}{259308 \, {\left (81 \, x^{6} + 54 \, x^{5} + 117 \, x^{4} + 72 \, x^{3} + 48 \, x^{2} + 24 \, x + 4\right )}} \] Input:
integrate(1/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="fricas")
Output:
-1/259308*(565*(81*x^6 + 54*x^5 + 117*x^4 + 72*x^3 + 48*x^2 + 24*x + 4)*we ierstrassPInverse(-68/27, -440/729, x + 1/9) + 16830*(81*x^6 + 54*x^5 + 11 7*x^4 + 72*x^3 + 48*x^2 + 24*x + 4)*weierstrassZeta(-68/27, -440/729, weie rstrassPInverse(-68/27, -440/729, x + 1/9)) + 9*(50490*x^5 + 29745*x^4 + 5 9970*x^3 + 32163*x^2 + 16854*x + 6850)*sqrt(9*x^3 + 3*x^2 + 6*x + 2))/(81* x^6 + 54*x^5 + 117*x^4 + 72*x^3 + 48*x^2 + 24*x + 4)
\[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int \frac {1}{\left (9 x^{3} + 3 x^{2} + 6 x + 2\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(9*x**3+3*x**2+6*x+2)**(5/2),x)
Output:
Integral((9*x**3 + 3*x**2 + 6*x + 2)**(-5/2), x)
\[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="maxima")
Output:
integrate((9*x^3 + 3*x^2 + 6*x + 2)^(-5/2), x)
\[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (9 \, x^{3} + 3 \, x^{2} + 6 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(9*x^3+3*x^2+6*x+2)^(5/2),x, algorithm="giac")
Output:
integrate((9*x^3 + 3*x^2 + 6*x + 2)^(-5/2), x)
Timed out. \[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int \frac {1}{{\left (9\,x^3+3\,x^2+6\,x+2\right )}^{5/2}} \,d x \] Input:
int(1/(6*x + 3*x^2 + 9*x^3 + 2)^(5/2),x)
Output:
int(1/(6*x + 3*x^2 + 9*x^3 + 2)^(5/2), x)
\[ \int \frac {1}{\left (2+6 x+3 x^2+9 x^3\right )^{5/2}} \, dx=\int \frac {\sqrt {9 x^{3}+3 x^{2}+6 x +2}}{729 x^{9}+729 x^{8}+1701 x^{7}+1485 x^{6}+1458 x^{5}+1026 x^{4}+540 x^{3}+252 x^{2}+72 x +8}d x \] Input:
int(1/(9*x^3+3*x^2+6*x+2)^(5/2),x)
Output:
int(sqrt(9*x**3 + 3*x**2 + 6*x + 2)/(729*x**9 + 729*x**8 + 1701*x**7 + 148 5*x**6 + 1458*x**5 + 1026*x**4 + 540*x**3 + 252*x**2 + 72*x + 8),x)