Integrand size = 31, antiderivative size = 732 \[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=-\frac {\left (9 B c d^3-24 A c d^2 e+84 a B d e^2+64 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 c e^2 x}+\frac {\left (3 B c d^2+56 A c d e-12 a B e^2\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{192 c e}+\frac {1}{48} (9 B d+8 A e) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {1}{8} B e x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (9 B c d^3-24 A c d^2 e+84 a B d e^2+64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (3 B c d^3+248 A c d^2 e+108 a B d e^2+64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 \sqrt {c} e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (8 A c d e \left (c d^2+12 a e^2\right )-B \left (3 c^2 d^4-24 a c d^2 e^2-16 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 c e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/384*(64*A*a*e^3-24*A*c*d^2*e+84*B*a*d*e^2+9*B*c*d^3)*(e*x^2+d)^(1/2)*(- c*x^4+a)^(1/2)/c/e^2/x+1/192*(56*A*c*d*e-12*B*a*e^2+3*B*c*d^2)*x*(e*x^2+d) ^(1/2)*(-c*x^4+a)^(1/2)/c/e+1/48*(8*A*e+9*B*d)*x^3*(e*x^2+d)^(1/2)*(-c*x^4 +a)^(1/2)+1/8*B*e*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/384*(d+a^(1/2)*e/ c^(1/2))*(64*A*a*e^3-24*A*c*d^2*e+84*B*a*d*e^2+9*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^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/384*a^(1/2)*(64*A*a*e^3+248*A*c* d^2*e+108*B*a*d*e^2+3*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))/c^(1/2)/e/(e*x^2+d)^(1/2 )/(-c*x^4+a)^(1/2)+1/128*(8*A*c*d*e*(12*a*e^2+c*d^2)-B*(-16*a^2*e^4-24*a*c *d^2*e^2+3*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))/c/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a )^(1/2)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx \] Input:
Integrate[(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4],x]
Output:
Integrate[(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4], 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.
| \(\displaystyle \int \sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}dx\) |
Input:
Int[(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4],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]
\[\int \left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="fricas" )
Output:
integral((B*e*x^4 + (B*d + A*e)*x^2 + A*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d ), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int \left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="maxima" )
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int \left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:
int((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2),x)
Output:
int((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\frac {-12 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b \,e^{2} x +56 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d e x +32 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,e^{2} x^{3}+3 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x +36 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d e \,x^{3}+24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,e^{2} x^{5}+64 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,e^{3}+84 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c d \,e^{2}-24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{2} e +9 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{3}+24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b \,e^{3}+176 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c d \,e^{2}+78 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{2} e +12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b d \,e^{2}+136 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,d^{2} e -3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{3}}{192 c e} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
Output:
( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2*x + 56*sqrt(d + e*x**2)* sqrt(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 + 3*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*x + 36*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 + 64*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 + 84*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 - 24*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 + 9*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 + 24*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*b*e**3 + 176*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 + 78*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 + 12*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 + 136*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 - 3*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*c*e)