Integrand size = 31, antiderivative size = 732 \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {\left (105 B c d^3-120 A c d^2 e-188 a B d e^2+256 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 e^4 x}-\frac {5 \left (7 B c d^2-8 A c d e-12 a B e^2\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{192 e^3}+\frac {c (7 B d-8 A e) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 e^2}-\frac {B c x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 e}+\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (105 B c d^3-120 A c d^2 e-188 a B d e^2+256 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \sqrt {c} \left (35 B c d^3-40 A c d^2 e-68 a B d e^2-128 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (8 A c d e \left (5 c d^2-12 a e^2\right )-B \left (35 c^2 d^4-72 a c d^2 e^2+48 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/384*(256*A*a*e^3-120*A*c*d^2*e-188*B*a*d*e^2+105*B*c*d^3)*(e*x^2+d)^(1/2 )*(-c*x^4+a)^(1/2)/e^4/x-5/192*(-8*A*c*d*e-12*B*a*e^2+7*B*c*d^2)*x*(e*x^2+ d)^(1/2)*(-c*x^4+a)^(1/2)/e^3+1/48*c*(-8*A*e+7*B*d)*x^3*(e*x^2+d)^(1/2)*(- c*x^4+a)^(1/2)/e^2-1/8*B*c*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e+1/384*c* (d+a^(1/2)*e/c^(1/2))*(256*A*a*e^3-120*A*c*d^2*e-188*B*a*d*e^2+105*B*c*d^3 )*(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))/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/384*a^(1/2)*c^ (1/2)*(-128*A*a*e^3-40*A*c*d^2*e-68*B*a*d*e^2+35*B*c*d^3)*(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 ))/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/128*(8*A*c*d*e*(-12*a*e^2+5*c*d^ 2)-B*(48*a^2*e^4-72*a*c*d^2*e^2+35*c^2*d^4))*(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))/e^4/(e* x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx \] Input:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/Sqrt[d + e*x^2],x]
Output:
Integrate[((A + B*x^2)*(a - c*x^4)^(3/2))/Sqrt[d + e*x^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.
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2\right )}{\sqrt {d+e x^2}}dx\) |
Input:
Int[((A + B*x^2)*(a - c*x^4)^(3/2))/Sqrt[d + e*x^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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\sqrt {e \,x^{2}+d}}d x\]
Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
Output:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\text {Timed out} \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="fricas" )
Output:
Timed out
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (a - c x^{4}\right )^{\frac {3}{2}}}{\sqrt {d + e x^{2}}}\, dx \] Input:
integrate((B*x**2+A)*(-c*x**4+a)**(3/2)/(e*x**2+d)**(1/2),x)
Output:
Integral((A + B*x**2)*(a - c*x**4)**(3/2)/sqrt(d + e*x**2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/sqrt(e*x^2 + d), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)/sqrt(e*x^2 + d), x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}}{\sqrt {e\,x^2+d}} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(1/2),x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(3/2))/(d + e*x^2)^(1/2), x)
\[ \int \frac {\left (A+B x^2\right ) \left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {60 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b \,e^{2} x +40 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d e x -32 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,e^{2} x^{3}-35 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x +28 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d e \,x^{3}-24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,e^{2} x^{5}-256 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,e^{3}+188 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c d \,e^{2}+120 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{2} e -105 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{3}+72 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b \,e^{3}+16 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c d \,e^{2}-14 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{2} e +192 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{3} e^{3}-60 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b d \,e^{2}-40 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,d^{2} e +35 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{3}}{192 e^{3}} \] Input:
int((B*x^2+A)*(-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
Output:
(60*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2*x + 40*sqrt(d + e*x**2)*sqr t(a - c*x**4)*a*c*d*e*x - 32*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e**2*x* *3 - 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*x + 28*sqrt(d + e*x**2) *sqrt(a - c*x**4)*b*c*d*e*x**3 - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c* e**2*x**5 - 256*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x* *2 - c*d*x**4 - c*e*x**6),x)*a**2*c*e**3 + 188*int((sqrt(d + e*x**2)*sqrt( a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d*e**2 + 120*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x* *4 - c*e*x**6),x)*a*c**2*d**2*e - 105*int((sqrt(d + e*x**2)*sqrt(a - c*x** 4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b*c**2*d**3 + 72*int((s qrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x* *6),x)*a**2*b*e**3 + 16*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*c*d*e**2 - 14*int((sqrt(d + e*x* *2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c *d**2*e + 192*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c* d*x**4 - c*e*x**6),x)*a**3*e**3 - 60*int((sqrt(d + e*x**2)*sqrt(a - c*x**4 ))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*b*d*e**2 - 40*int((sqrt( d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a* *2*c*d**2*e + 35*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d**3)/(192*e**3)