Integrand size = 46, antiderivative size = 196 \[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\frac {x \left (a+b x^2+c x^4\right )^{3/2}}{4 d e \left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )^2}+\frac {3 x \sqrt {a+b x^2+c x^4}}{8 d^2 e^2 \left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )}+\frac {3 \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{8 d^{5/2} e^{5/2} \sqrt {c d^2-b d e+a e^2}} \] Output:
1/4*x*(c*x^4+b*x^2+a)^(3/2)/d/e/(a*d*e+(a*e^2+c*d^2)*x^2+c*d*e*x^4)^2+3/8* x*(c*x^4+b*x^2+a)^(1/2)/d^2/e^2/(a*d*e+(a*e^2+c*d^2)*x^2+c*d*e*x^4)+3/8*ar ctan((a*e^2-b*d*e+c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^ (5/2)/e^(5/2)/(a*e^2-b*d*e+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.68 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\frac {\left (c d^2-a e^2\right )^3 \left (\frac {2 d e x \left (a+b x^2+c x^4\right ) \left (a e \left (5 d+3 e x^2\right )+d x^2 \left (3 c d+2 b e+5 c e x^2\right )\right )}{\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2}+\frac {3 i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}\right )}{16 \left (c d^3 e-a d e^3\right )^3 \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[((a - c*x^4)*(a + b*x^2 + c*x^4)^(3/2))/((a*e + c*d*x^2)^3*(d + e*x^2)^3),x]
Output:
((c*d^2 - a*e^2)^3*((2*d*e*x*(a + b*x^2 + c*x^4)*(a*e*(5*d + 3*e*x^2) + d* x^2*(3*c*d + 2*b*e + 5*c*e*x^2)))/((a*e + c*d*x^2)^2*(d + e*x^2)^2) + ((3* I)*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])] *Sqrt[1 + (2*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])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(2*a*e), I*ArcSinh[Sqr t[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])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSin h[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/(b + Sqrt[b^2 - 4*a*c])]))/(16*(c*d^3*e - a* d*e^3)^3*Sqrt[a + b*x^2 + c*x^4])
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.
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )}-\frac {a c^2 d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^3 \left (a e+c d x^2\right )^2}+\frac {a c^2 d \left (a+b x^2+c x^4\right )^{3/2}}{\left (c d^2-a e^2\right )^2 \left (a e+c d x^2\right )^3}+\frac {c e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c d e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^3}+\frac {e \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^3 \left (a e+c d x^2\right )^3}dx\) |
Input:
Int[((a - c*x^4)*(a + b*x^2 + c*x^4)^(3/2))/((a*e + c*d*x^2)^3*(d + e*x^2) ^3),x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.56 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (5 c d e \,x^{4}+3 a \,e^{2} x^{2}+2 b d e \,x^{2}+3 c \,d^{2} x^{2}+5 a d e \right ) x}{\left (c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e \right )^{2}}-\frac {3 \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}}{8 d^{2} e^{2}}\) | \(161\) |
| elliptic | \(\frac {\left (-\frac {32 \left (-\frac {5 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {2}}{256 d e \,x^{3}}-\frac {3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{256 d^{2} e^{2} x}\right )}{{\left (\frac {d e \left (c \,x^{4}+b \,x^{2}+a \right )}{x^{2}}+a \,e^{2}-b d e +c \,d^{2}\right )}^{2}}-\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{8 d^{2} e^{2} \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right ) \sqrt {2}}{2}\) | \(192\) |
| default | \(\text {Expression too large to display}\) | \(18352\) |
Input:
int((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x,method= _RETURNVERBOSE)
Output:
1/8/d^2/e^2*((c*x^4+b*x^2+a)^(1/2)*(5*c*d*e*x^4+3*a*e^2*x^2+2*b*d*e*x^2+3* c*d^2*x^2+5*a*d*e)*x/(c*d*e*x^4+a*e^2*x^2+c*d^2*x^2+a*d*e)^2-3/((a*e^2-b*d *e+c*d^2)*d*e)^(1/2)*arctan((c*x^4+b*x^2+a)^(1/2)/x*d*e/((a*e^2-b*d*e+c*d^ 2)*d*e)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (172) = 344\).
Time = 27.36 (sec) , antiderivative size = 1496, normalized size of antiderivative = 7.63 \[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x, algorithm="fricas")
Output:
[-1/32*(3*(c^2*d^2*e^2*x^8 + 2*(c^2*d^3*e + a*c*d*e^3)*x^6 + a^2*d^2*e^2 + (c^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4)*x^4 + 2*(a*c*d^3*e + a^2*d*e^3)*x^2)* sqrt(-c*d^3*e + b*d^2*e^2 - a*d*e^3)*log(-(c^2*d^2*e^2*x^8 - 2*(3*c^2*d^3* e - 4*b*c*d^2*e^2 + 3*a*c*d*e^3)*x^6 + a^2*d^2*e^2 + (c^2*d^4 - 8*b*c*d^3* e - 8*a*b*d*e^3 + a^2*e^4 + 4*(2*b^2 + a*c)*d^2*e^2)*x^4 - 2*(3*a*c*d^3*e - 4*a*b*d^2*e^2 + 3*a^2*d*e^3)*x^2 + 4*(c*d*e*x^5 + a*d*e*x - (c*d^2 - 2*b *d*e + a*e^2)*x^3)*sqrt(-c*d^3*e + b*d^2*e^2 - a*d*e^3)*sqrt(c*x^4 + b*x^2 + a))/(c^2*d^2*e^2*x^8 + 2*(c^2*d^3*e + a*c*d*e^3)*x^6 + a^2*d^2*e^2 + (c ^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4)*x^4 + 2*(a*c*d^3*e + a^2*d*e^3)*x^2)) - 4*(5*(c^2*d^4*e^2 - b*c*d^3*e^3 + a*c*d^2*e^4)*x^5 + (3*c^2*d^5*e - b*c*d^ 4*e^2 - a*b*d^2*e^4 + 3*a^2*d*e^5 - 2*(b^2 - 3*a*c)*d^3*e^3)*x^3 + 5*(a*c* d^4*e^2 - a*b*d^3*e^3 + a^2*d^2*e^4)*x)*sqrt(c*x^4 + b*x^2 + a))/(a^2*c*d^ 7*e^5 - a^2*b*d^6*e^6 + a^3*d^5*e^7 + (c^3*d^7*e^5 - b*c^2*d^6*e^6 + a*c^2 *d^5*e^7)*x^8 + 2*(c^3*d^8*e^4 - b*c^2*d^7*e^5 + 2*a*c^2*d^6*e^6 - a*b*c*d ^5*e^7 + a^2*c*d^4*e^8)*x^6 + (c^3*d^9*e^3 - b*c^2*d^8*e^4 + 5*a*c^2*d^7*e ^5 - 4*a*b*c*d^6*e^6 + 5*a^2*c*d^5*e^7 - a^2*b*d^4*e^8 + a^3*d^3*e^9)*x^4 + 2*(a*c^2*d^8*e^4 - a*b*c*d^7*e^5 + 2*a^2*c*d^6*e^6 - a^2*b*d^5*e^7 + a^3 *d^4*e^8)*x^2), 1/16*(3*(c^2*d^2*e^2*x^8 + 2*(c^2*d^3*e + a*c*d*e^3)*x^6 + a^2*d^2*e^2 + (c^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4)*x^4 + 2*(a*c*d^3*e + a^ 2*d*e^3)*x^2)*sqrt(c*d^3*e - b*d^2*e^2 + a*d*e^3)*arctan(2*sqrt(c*d^3*e...
Timed out. \[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((-c*x**4+a)*(c*x**4+b*x**2+a)**(3/2)/(c*d*x**2+a*e)**3/(e*x**2+d )**3,x)
Output:
Timed out
\[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\int { -\frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (c x^{4} - a\right )}}{{\left (c d x^{2} + a e\right )}^{3} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x, algorithm="maxima")
Output:
-integrate((c*x^4 + b*x^2 + a)^(3/2)*(c*x^4 - a)/((c*d*x^2 + a*e)^3*(e*x^2 + d)^3), x)
\[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\int { -\frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (c x^{4} - a\right )}}{{\left (c d x^{2} + a e\right )}^{3} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate(-(c*x^4 + b*x^2 + a)^(3/2)*(c*x^4 - a)/((c*d*x^2 + a*e)^3*(e*x^2 + d)^3), x)
Timed out. \[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\int \frac {\left (a-c\,x^4\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (e\,x^2+d\right )}^3\,{\left (c\,d\,x^2+a\,e\right )}^3} \,d x \] Input:
int(((a - c*x^4)*(a + b*x^2 + c*x^4)^(3/2))/((d + e*x^2)^3*(a*e + c*d*x^2) ^3),x)
Output:
int(((a - c*x^4)*(a + b*x^2 + c*x^4)^(3/2))/((d + e*x^2)^3*(a*e + c*d*x^2) ^3), x)
\[ \int \frac {\left (a-c x^4\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (a e+c d x^2\right )^3 \left (d+e x^2\right )^3} \, dx=\int \frac {\left (-c \,x^{4}+a \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (c d \,x^{2}+a e \right )^{3} \left (e \,x^{2}+d \right )^{3}}d x \] Input:
int((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x)
Output:
int((-c*x^4+a)*(c*x^4+b*x^2+a)^(3/2)/(c*d*x^2+a*e)^3/(e*x^2+d)^3,x)