Integrand size = 34, antiderivative size = 702 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\frac {(d+4 e x) \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2}}{28 e}-\frac {3 d^2 \left (d^4+512 a e^3\right ) (d+4 e x) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{280 e^2 \left (\sqrt {5 d^4+256 a e^3}+(d+4 e x)^2\right )}+\frac {(d+4 e x) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \left (43 d^4+1280 a e^3-9 d^2 (d+4 e x)^2\right )}{2240 e^2}+\frac {3 d^2 \left (5 d^4+256 a e^3\right )^{3/4} \left (d^4+512 a e^3\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right ) \sqrt {\frac {5 d^4+256 a e^3-6 d^2 (d+4 e x)^2+(d+4 e x)^4}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} E\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{8960 e^3 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac {\left (5 d^4+256 a e^3\right )^{3/4} \left (4 \left (d^4+80 a e^3\right ) \sqrt {5 d^4+256 a e^3}-3 d^2 \left (d^4+512 a e^3\right )\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right ) \sqrt {\frac {5 d^4+256 a e^3-6 d^2 (d+4 e x)^2+(d+4 e x)^4}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{17920 e^3 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \] Output:
1/28*(4*e*x+d)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2)/e-3/280*d^2*(51 2*a*e^3+d^4)*(4*e*x+d)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)/e^2/((2 56*a*e^3+5*d^4)^(1/2)+(4*e*x+d)^2)+1/2240*(4*e*x+d)*(8*e^3*x^4+8*d*e^2*x^3 -d^3*x+8*a*e^2)^(1/2)*(43*d^4+1280*a*e^3-9*d^2*(4*e*x+d)^2)/e^2+3/8960*d^2 *(256*a*e^3+5*d^4)^(3/4)*(512*a*e^3+d^4)*(1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^ (1/2))*((5*d^4+256*a*e^3-6*d^2*(4*e*x+d)^2+(4*e*x+d)^4)/(256*a*e^3+5*d^4)/ (1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^(1/2))^2)^(1/2)*EllipticE(sin(2*arctan((4 *e*x+d)/(256*a*e^3+5*d^4)^(1/4))),1/2*(2+6*d^2/(256*a*e^3+5*d^4)^(1/2))^(1 /2))/e^3/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)+1/17920*(256*a*e^3+5* d^4)^(3/4)*(4*(80*a*e^3+d^4)*(256*a*e^3+5*d^4)^(1/2)-3*d^2*(512*a*e^3+d^4) )*(1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^(1/2))*((5*d^4+256*a*e^3-6*d^2*(4*e*x+d )^2+(4*e*x+d)^4)/(256*a*e^3+5*d^4)/(1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^(1/2)) ^2)^(1/2)*InverseJacobiAM(2*arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4)),1/2* (2+6*d^2/(256*a*e^3+5*d^4)^(1/2))^(1/2))/e^3/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+ 8*a*e^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(13510\) vs. \(2(702)=1404\).
Time = 16.24 (sec) , antiderivative size = 13510, normalized size of antiderivative = 19.25 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\text {Result too large to show} \] Input:
Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(3/2),x]
Output:
Result too large to show
Time = 0.96 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.35, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2458, 1404, 27, 1490, 27, 1511, 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.
| \(\displaystyle \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}d\left (\frac {d}{4 e}+x\right )\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int \frac {\left (\frac {5 d^4}{e}-48 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 a e^2\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}{64 \sqrt {2}}d\left (\frac {d}{4 e}+x\right )+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \left (\frac {5 d^4}{e}-48 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 a e^2\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}d\left (\frac {d}{4 e}+x\right )}{448 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {8192 e \left (\left (d^4+80 a e^3\right ) \left (5 d^4+256 a e^3\right )-12 d^2 e^2 \left (d^4+512 a e^3\right ) \left (\frac {d}{4 e}+x\right )^2\right )}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{3840 e^3}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4} \left (1280 a e^3+43 d^4-144 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{15 e}\right )}{448 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {32 \int \frac {\left (d^4+80 a e^3\right ) \left (5 d^4+256 a e^3\right )-12 d^2 e^2 \left (d^4+512 a e^3\right ) \left (\frac {d}{4 e}+x\right )^2}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{15 e^2}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4} \left (1280 a e^3+43 d^4-144 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{15 e}\right )}{448 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3 \left (\frac {32 \left (\frac {3}{4} d^2 \sqrt {256 a e^3+5 d^4} \left (512 a e^3+d^4\right ) \int \frac {1-\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )+\frac {1}{4} \sqrt {256 a e^3+5 d^4} \left (4 \left (80 a e^3+d^4\right ) \sqrt {256 a e^3+5 d^4}-3 d^2 \left (512 a e^3+d^4\right )\right ) \int \frac {1}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )\right )}{15 e^2}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4} \left (1280 a e^3+43 d^4-144 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{15 e}\right )}{448 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3 \left (\frac {32 \left (\frac {3}{4} d^2 \sqrt {256 a e^3+5 d^4} \left (512 a e^3+d^4\right ) \int \frac {1-\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )+\frac {\left (256 a e^3+5 d^4\right )^{3/4} \left (4 \left (80 a e^3+d^4\right ) \sqrt {256 a e^3+5 d^4}-3 d^2 \left (512 a e^3+d^4\right )\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {256 a e^3+5 d^4}}+1\right ) \sqrt {\frac {256 a e^3+5 d^4-96 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4}{\left (256 a e^3+5 d^4\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {256 a e^3+5 d^4}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{32 e \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}\right )}{15 e^2}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4} \left (1280 a e^3+43 d^4-144 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2\right )}{15 e}\right )}{448 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \left (256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^{3/2}}{896 \sqrt {2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\left (\frac {d}{4 e}+x\right ) \left (\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2\right )^{3/2}}{896 \sqrt {2}}+\frac {3 \left (\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2} \left (43 d^4-144 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+1280 a e^3\right )}{15 e}+\frac {32 \left (\frac {3}{4} \sqrt {5 d^4+256 a e^3} \left (d^4+512 a e^3\right ) \left (\frac {\sqrt [4]{5 d^4+256 a e^3} \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} E\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{4 e \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}-\frac {e \left (\frac {d}{4 e}+x\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )}\right ) d^2+\frac {\left (5 d^4+256 a e^3\right )^{3/4} \left (4 \left (d^4+80 a e^3\right ) \sqrt {5 d^4+256 a e^3}-3 d^2 \left (d^4+512 a e^3\right )\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{32 e \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}\right )}{15 e^2}\right )}{448 \sqrt {2}}\) |
Input:
Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(3/2),x]
Output:
((d/(4*e) + x)*((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3 *(d/(4*e) + x)^4)^(3/2))/(896*Sqrt[2]) + (3*(((d/(4*e) + x)*(43*d^4 + 1280 *a*e^3 - 144*d^2*e^2*(d/(4*e) + x)^2)*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2* e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])/(15*e) + (32*((3*d^2*Sqrt[5* d^4 + 256*a*e^3]*(d^4 + 512*a*e^3)*(-((e*(d/(4*e) + x)*Sqrt[(5*d^4)/e + 25 6*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])/((5*d^4 + 2 56*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3]))) + ((5*d ^4 + 256*a*e^3)^(1/4)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3 ])*Sqrt[(5*d^4 + 256*a*e^3 - 96*d^2*e^2*(d/(4*e) + x)^2 + 256*e^4*(d/(4*e) + x)^4)/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 2 56*a*e^3])^2)]*EllipticE[2*ArcTan[(4*e*(d/(4*e) + x))/(5*d^4 + 256*a*e^3)^ (1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a*e^3])/2])/(4*e*Sqrt[(5*d^4)/e + 25 6*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])))/4 + ((5*d ^4 + 256*a*e^3)^(3/4)*(4*(d^4 + 80*a*e^3)*Sqrt[5*d^4 + 256*a*e^3] - 3*d^2* (d^4 + 512*a*e^3))*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])* Sqrt[(5*d^4 + 256*a*e^3 - 96*d^2*e^2*(d/(4*e) + x)^2 + 256*e^4*(d/(4*e) + x)^4)/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256* a*e^3])^2)]*EllipticF[2*ArcTan[(4*e*(d/(4*e) + x))/(5*d^4 + 256*a*e^3)^(1/ 4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a*e^3])/2])/(32*e*Sqrt[(5*d^4)/e + 256* a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])))/(15*e^2)...
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. \(8240\) vs. \(2(655)=1310\).
Time = 11.79 (sec) , antiderivative size = 8241, normalized size of antiderivative = 11.74
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(8241\) |
| elliptic | \(\text {Expression too large to display}\) | \(8241\) |
| risch | \(\text {Expression too large to display}\) | \(11353\) |
Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
\[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\int { {\left (8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2),x, algorithm="fricas ")
Output:
integral((8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2)^(3/2), x)
\[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\int \left (8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**(3/2),x)
Output:
Integral((8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)**(3/2), x)
\[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\int { {\left (8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2),x, algorithm="maxima ")
Output:
integrate((8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2)^(3/2), x)
\[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\int { {\left (8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2),x, algorithm="giac")
Output:
integrate((8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2)^(3/2), x)
Timed out. \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx=\int {\left (-d^3\,x+8\,d\,e^2\,x^3+8\,e^3\,x^4+8\,a\,e^2\right )}^{3/2} \,d x \] Input:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^(3/2),x)
Output:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^(3/2), x)
\[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(3/2),x)
Output:
(1920*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*a*d*e**4 + 768 0*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*a*e**5*x + 34*sqrt (8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d**5*e - 16*sqrt(8*a*e** 2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d**4*e**2*x - 752*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d**3*e**3*x**2 + 64*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d**2*e**4*x**3 + 3200*sqrt(8*a*e** 2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d*e**5*x**4 + 2560*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*e**6*x**5 - 1536*sqrt(e)*sqrt(2)* log( - 2*sqrt(e)*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqr t(2) + d**2 - 4*d*e*x - 8*e**2*x**2)*a*d**3*e**3 - 3*sqrt(e)*sqrt(2)*log( - 2*sqrt(e)*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqrt(2) + d**2 - 4*d*e*x - 8*e**2*x**2)*d**7 - 1536*sqrt(e)*sqrt(2)*log(2*sqrt(e)* sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqrt(2) + d**2 - 4*d *e*x - 8*e**2*x**2)*a*d**3*e**3 - 3*sqrt(e)*sqrt(2)*log(2*sqrt(e)*sqrt(8*a *e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqrt(2) + d**2 - 4*d*e*x - 8 *e**2*x**2)*d**7 + 81920*int(sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e* *3*x**4)/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4),x)*a**2*e**7 + 1088*int(sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4),x)*a*d**4*e**4 + 17*int(sqrt(8*a*e* *2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)/(8*a*e**2 - d**3*x + 8*d*e**...