Integrand size = 31, antiderivative size = 683 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\frac {(c+d x) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}}{7 d}-\frac {16 c^3 \left (c^3+8 a d^2\right ) (c+d x) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^3 \left (\sqrt {c} \sqrt {c^3+4 a d^2}+(c+d x)^2\right )}+\frac {2 c (c+d x) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (7 c^3+20 a d^2-3 c (c+d x)^2\right )}{35 d^3}+\frac {16 c^{13/4} \left (c^3+4 a d^2\right )^{3/4} \left (c^3+8 a d^2\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) E\left (2 \arctan \left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{35 d^5 \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {8 c^{7/4} \left (c^3+4 a d^2\right )^{3/4} \left (\sqrt {c^3+4 a d^2} \left (c^3+5 a d^2\right )-c^{3/2} \left (c^3+8 a d^2\right )\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{35 d^5 \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \] Output:
1/7*(d*x+c)*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2)/d-16/35*c^3*(8*a*d^2 +c^3)*(d*x+c)*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)/d^3/(c^(1/2)*(4*a* d^2+c^3)^(1/2)+(d*x+c)^2)+2/35*c*(d*x+c)*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a* c)^(1/2)*(7*c^3+20*a*d^2-3*c*(d*x+c)^2)/d^3+16/35*c^(13/4)*(4*a*d^2+c^3)^( 3/4)*(8*a*d^2+c^3)*(d^2*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)/(4*a*d^2+c^3)/ (c^(1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))^2)^(1/2)*(c^(1/2)+(d*x+c)^2/(4*a*d ^2+c^3)^(1/2))*EllipticE(sin(2*arctan((d*x+c)/c^(1/4)/(4*a*d^2+c^3)^(1/4)) ),1/2*(2+2*c^(3/2)/(4*a*d^2+c^3)^(1/2))^(1/2))/d^5/(d^2*x^4+4*c*d*x^3+4*c^ 2*x^2+4*a*c)^(1/2)+8/35*c^(7/4)*(4*a*d^2+c^3)^(3/4)*((4*a*d^2+c^3)^(1/2)*( 5*a*d^2+c^3)-c^(3/2)*(8*a*d^2+c^3))*(d^2*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a* c)/(4*a*d^2+c^3)/(c^(1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))^2)^(1/2)*(c^(1/2) +(d*x+c)^2/(4*a*d^2+c^3)^(1/2))*InverseJacobiAM(2*arctan((d*x+c)/c^(1/4)/( 4*a*d^2+c^3)^(1/4)),1/2*(2+2*c^(3/2)/(4*a*d^2+c^3)^(1/2))^(1/2))/d^5/(d^2* x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)
Result contains complex when optimal does not.
Time = 16.22 (sec) , antiderivative size = 10468, normalized size of antiderivative = 15.33 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\text {Result too large to show} \] Input:
Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2),x]
Output:
Result too large to show
Time = 1.02 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2458, 1404, 27, 1490, 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.
| \(\displaystyle \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int 2 c \left (\frac {c^3}{d^2}-\left (\frac {c}{d}+x\right )^2 c+4 a\right ) \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}d\left (\frac {c}{d}+x\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} c \int \left (\frac {c^3}{d^2}-\left (\frac {c}{d}+x\right )^2 c+4 a\right ) \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}d\left (\frac {c}{d}+x\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {6}{7} c \left (\frac {\int \frac {8 c \left (\left (c^3+4 a d^2\right ) \left (c^3+5 a d^2\right )-c d^2 \left (c^3+8 a d^2\right ) \left (\frac {c}{d}+x\right )^2\right )}{d^2 \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{15 d^2}+\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac {c}{d}+x\right )^2\right )}{15 d^2}\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} c \left (\frac {8 c \int \frac {\left (c^3+4 a d^2\right ) \left (c^3+5 a d^2\right )-c d^2 \left (c^3+8 a d^2\right ) \left (\frac {c}{d}+x\right )^2}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{15 d^4}+\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac {c}{d}+x\right )^2\right )}{15 d^2}\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {6}{7} c \left (\frac {8 c \left (c^{3/2} \sqrt {4 a d^2+c^3} \left (8 a d^2+c^3\right ) \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {c} \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )-\sqrt {4 a d^2+c^3} \left (8 a c^{3/2} d^2-\sqrt {4 a d^2+c^3} \left (5 a d^2+c^3\right )+c^{9/2}\right ) \int \frac {1}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )\right )}{15 d^4}+\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac {c}{d}+x\right )^2\right )}{15 d^2}\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} c \left (\frac {8 c \left (c \sqrt {4 a d^2+c^3} \left (8 a d^2+c^3\right ) \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )-\sqrt {4 a d^2+c^3} \left (8 a c^{3/2} d^2-\sqrt {4 a d^2+c^3} \left (5 a d^2+c^3\right )+c^{9/2}\right ) \int \frac {1}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )\right )}{15 d^4}+\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac {c}{d}+x\right )^2\right )}{15 d^2}\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {6}{7} c \left (\frac {8 c \left (c \sqrt {4 a d^2+c^3} \left (8 a d^2+c^3\right ) \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )-\frac {\left (4 a d^2+c^3\right )^{3/4} \left (8 a c^{3/2} d^2-\sqrt {4 a d^2+c^3} \left (5 a d^2+c^3\right )+c^{9/2}\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}}\right )}{15 d^4}+\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac {c}{d}+x\right )^2\right )}{15 d^2}\right )+\frac {1}{7} \left (\frac {c}{d}+x\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{7} \left (\frac {c}{d}+x\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )^{3/2}+\frac {6}{7} c \left (\frac {\left (\frac {c}{d}+x\right ) \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )} \left (7 c^3-3 d^2 \left (\frac {c}{d}+x\right )^2 c+20 a d^2\right )}{15 d^2}+\frac {8 c \left (c \sqrt {c^3+4 a d^2} \left (c^3+8 a d^2\right ) \left (\frac {\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}{\left (c^3+4 a d^2\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{d \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}-\frac {\left (\frac {c}{d}+x\right ) \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}{\left (\frac {c^3}{d^2}+4 a\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}+\sqrt {c}\right )}\right )-\frac {\left (c^3+4 a d^2\right )^{3/4} \left (c^{9/2}+8 a d^2 c^{3/2}-\sqrt {c^3+4 a d^2} \left (c^3+5 a d^2\right )\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}{\left (c^3+4 a d^2\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}\right )}{15 d^4}\right )\) |
Input:
Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2),x]
Output:
((c/d + x)*(c*(4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4)^(3/2) )/7 + (6*c*(((c/d + x)*(7*c^3 + 20*a*d^2 - 3*c*d^2*(c/d + x)^2)*Sqrt[c*(4* a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4])/(15*d^2) + (8*c*(c*Sq rt[c^3 + 4*a*d^2]*(c^3 + 8*a*d^2)*(-(((c/d + x)*Sqrt[c*(4*a + c^3/d^2) - 2 *c^2*(c/d + x)^2 + d^2*(c/d + x)^4])/((4*a + c^3/d^2)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2]))) + (c^(1/4)*(c^3 + 4*a*d^2)^(1/4)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*Sqrt[(d^2*(c*(4*a + c^3/d^2) - 2* c^2*(c/d + x)^2 + d^2*(c/d + x)^4))/((c^3 + 4*a*d^2)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])^2)]*EllipticE[2*ArcTan[(d*(c/d + x))/(c^(1/4) *(c^3 + 4*a*d^2)^(1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/(d*Sqrt[c* (4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4])) - ((c^3 + 4*a*d^2 )^(3/4)*(c^(9/2) + 8*a*c^(3/2)*d^2 - Sqrt[c^3 + 4*a*d^2]*(c^3 + 5*a*d^2))* (Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*Sqrt[(d^2*(c*(4*a + c^3/ d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4))/((c^3 + 4*a*d^2)*(Sqrt[c] + ( d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])^2)]*EllipticF[2*ArcTan[(d*(c/d + x)) /(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/( 2*c^(1/4)*d*Sqrt[c*(4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4]) ))/(15*d^4)))/7
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 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b *x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1)) Int[(2*a + b*x^2)*( a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* c, 0] && GtQ[p, 0] && IntegerQ[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)*((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_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(5228\) vs. \(2(614)=1228\).
Time = 8.95 (sec) , antiderivative size = 5229, normalized size of antiderivative = 7.66
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(5229\) |
| elliptic | \(\text {Expression too large to display}\) | \(5229\) |
| risch | \(\text {Expression too large to display}\) | \(6015\) |
Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
\[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\int { {\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2),x, algorithm="fricas")
Output:
integral((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(3/2), x)
\[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\int \left (4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**(3/2),x)
Output:
Integral((4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)**(3/2), x)
\[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\int { {\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2),x, algorithm="maxima")
Output:
integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(3/2), x)
\[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\int { {\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2),x, algorithm="giac")
Output:
integrate((d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c)^(3/2), x)
Timed out. \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\int {\left (4\,c^2\,x^2+4\,c\,d\,x^3+4\,a\,c+d^2\,x^4\right )}^{3/2} \,d x \] Input:
int((4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^(3/2),x)
Output:
int((4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^(3/2), x)
\[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx=\frac {116 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, a \,c^{2} d^{2}+180 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, a c \,d^{3} x +16 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c^{5}-12 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c^{4} d x +6 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c^{3} d^{2} x^{2}+102 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c^{2} d^{3} x^{3}+75 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c \,d^{4} x^{4}+15 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, d^{5} x^{5}+960 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) a^{2} c^{2} d^{3}+48 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) a \,c^{5} d +128 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x^{3}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) a \,c^{2} d^{4}+16 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x^{3}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) c^{5} d^{2}-512 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) a \,c^{4} d^{2}-64 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) c^{7}}{105 d^{3}} \] Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(3/2),x)
Output:
(116*sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*a*c**2*d**2 + 180* sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*a*c*d**3*x + 16*sqrt(4* a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*c**5 - 12*sqrt(4*a*c + 4*c**2* x**2 + 4*c*d*x**3 + d**2*x**4)*c**4*d*x + 6*sqrt(4*a*c + 4*c**2*x**2 + 4*c *d*x**3 + d**2*x**4)*c**3*d**2*x**2 + 102*sqrt(4*a*c + 4*c**2*x**2 + 4*c*d *x**3 + d**2*x**4)*c**2*d**3*x**3 + 75*sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x* *3 + d**2*x**4)*c*d**4*x**4 + 15*sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d **2*x**4)*d**5*x**5 + 960*int(sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2 *x**4)/(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4),x)*a**2*c**2*d**3 + 48*int(sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)/(4*a*c + 4*c**2* x**2 + 4*c*d*x**3 + d**2*x**4),x)*a*c**5*d + 128*int((sqrt(4*a*c + 4*c**2* x**2 + 4*c*d*x**3 + d**2*x**4)*x**3)/(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d **2*x**4),x)*a*c**2*d**4 + 16*int((sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*x**3)/(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4),x)*c**5*d **2 - 512*int((sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*x)/(4*a* c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4),x)*a*c**4*d**2 - 64*int((sqrt(4* a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*x)/(4*a*c + 4*c**2*x**2 + 4*c* d*x**3 + d**2*x**4),x)*c**7)/(105*d**3)