Integrand size = 31, antiderivative size = 568 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{2 a \sqrt {a-c x^4}}+\frac {(B d+A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c x}+\frac {B e x \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c}+\frac {(B d+A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (A c d^2-2 a B d e-a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {B e^2 \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{c \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/2*x*(B*x^2+A)*(e*x^2+d)^(3/2)/a/(-c*x^4+a)^(1/2)+1/2*(A*e+B*d)*(e*x^2+d) ^(1/2)*(-c*x^4+a)^(1/2)/a/c/x+1/2*B*e*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a /c+1/2*(A*e+B*d)*(d+a^(1/2)*e/c^(1/2))*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x ^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2*(1-a^(1/2)/c^(1/2)/x^ 2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/a/(e*x^2+d)^(1/2 )/(-c*x^4+a)^(1/2)+1/2*(-A*a*e^2+A*c*d^2-2*B*a*d*e)*(1-a/c/x^4)^(1/2)*x^3* (a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticF(1/2*(1-a^(1/ 2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/a^( 3/2)/c^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-B*e^2*(1-a/c/x^4)^(1/2)*x^3* (a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a^(1 /2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/ c/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(3/2),x]
Output:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(3/2), x]
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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}}dx\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(3/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas" )
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)/(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(d + e*x**2)**(3/2)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(3/2),x)
Output:
int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e**2*x + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*d*e*x + int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a**2*d + a **2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)* a*b*c*d*e**2 - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a**2*d + a**2 *e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*b*c **2*d*e**2*x**4 - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x )*a**3*e**3 - 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a** 2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a* *2*b*d*e**2 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a** 2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a* *2*c*d**2*e + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a** 2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a* *2*c*e**3*x**4 + int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a* *2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a *b*c*d**3 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a**2* e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)*a*b* c*d*e**2*x**4 - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2*d + a **2*e*x**2 - 2*a*c*d*x**4 - 2*a*c*e*x**6 + c**2*d*x**8 + c**2*e*x**10),x)* a*c**2*d**2*e*x**4 - int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a**2...