Integrand size = 43, antiderivative size = 416 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {B x \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 b d f}+\frac {\sqrt {a} (2 a B d f-b (3 B d e+2 B c f-3 A d f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f^2 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} \left (a B c d f^2+3 A b d f (d e+c f)-b B \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} e (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-1/3*B*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f+1/3*a^(1/2)*(2*a*B*d*f-b*( -3*A*d*f+2*B*c*f+3*B*d*e))*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^( 1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(3/2)/d^2/f^2/(-b*x^2+a)^(1/2)/(1+d*x^2 /c)^(1/2)-1/3*a^(1/2)*(a*B*c*d*f^2+3*A*b*d*f*(c*f+d*e)-b*B*(2*c^2*f^2+3*c* d*e*f+3*d^2*e^2))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/ a^(1/2),(-a*d/b/c)^(1/2))/b^(3/2)/d^2/f^3/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2) -a^(1/2)*e*(-A*f+B*e)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/ 2)*x/a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/f^3/(-b*x^2+a)^(1/2)/(d*x^ 2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.93 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.87 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {-\frac {b}{a}} B d f^2 x \left (a-b x^2\right ) \left (c+d x^2\right )-i c f (2 a B d f+b (-3 B d e-2 B c f+3 A d f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i \left (a B c d f^2+3 A b d f (d e+c f)-b B \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+3 i b d^2 e (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 b \sqrt {-\frac {b}{a}} d^2 f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^4*(A + B*x^2))/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)), x]
Output:
(-(Sqrt[-(b/a)]*B*d*f^2*x*(a - b*x^2)*(c + d*x^2)) - I*c*f*(2*a*B*d*f + b* (-3*B*d*e - 2*B*c*f + 3*A*d*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El lipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I*(a*B*c*d*f^2 + 3*A* b*d*f*(d*e + c*f) - b*B*(3*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2))*Sqrt[1 - (b*x ^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b *c))] + (3*I)*b*d^2*e*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(3* b*Sqrt[-(b/a)]*d^2*f^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
Time = 2.38 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {A e^2 f-B e^3}{f^3 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {e (B e-A f)}{f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {x^2 (B e-A f)}{f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {B x^4}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (B e-A f) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d f^2 \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}+\frac {\sqrt {a} B c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (2 b c-a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {B x \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 b d f}\) |
Input:
Int[(x^4*(A + B*x^2))/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
-1/3*(B*x*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d*f) - (2*Sqrt[a]*B*(b*c - a *d)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[ a]], -((a*d)/(b*c))])/(3*b^(3/2)*d^2*f*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c] ) - (Sqrt[a]*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[Arc Sin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f^2*Sqrt[a - b*x^2]* Sqrt[1 + (d*x^2)/c]) + (Sqrt[a]*B*c*(2*b*c - a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt [1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(3 *b^(3/2)*d^2*f*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[a]*e*(B*e - A*f)*S qrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a ]], -((a*d)/(b*c))])/(Sqrt[b]*f^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt [a]*c*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin [(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f^2*Sqrt[a - b*x^2]*Sqr t[c + d*x^2]) - (Sqrt[a]*e*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2 )/c]*EllipticPi[-((a*f)/(b*e)), ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c) )])/(Sqrt[b]*f^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 10.56 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {B x \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b d f}+\frac {\left (-\frac {\left (3 A b d f +2 a B d f -2 B b c f -3 b B d e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}-\frac {\left (3 A b d e f -B a c \,f^{2}-3 b B d \,e^{2}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {3 b d e \left (A f -B e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b d f \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(465\) |
| default | \(-\frac {\left (-B \sqrt {\frac {b}{a}}\, b \,d^{2} f^{2} x^{5}+B \sqrt {\frac {b}{a}}\, a \,d^{2} f^{2} x^{3}-B \sqrt {\frac {b}{a}}\, b c d \,f^{2} x^{3}+3 A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d \,f^{2}+3 A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,d^{2} e f -3 A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d \,f^{2}-3 A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b \,d^{2} e f +B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d \,f^{2}-2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f^{2}-3 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d e f -3 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,d^{2} e^{2}-2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d \,f^{2}+2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f^{2}+3 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d e f +3 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b \,d^{2} e^{2}+B \sqrt {\frac {b}{a}}\, a c d \,f^{2} x \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 d^{2} b \sqrt {\frac {b}{a}}\, f^{3} \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(864\) |
| elliptic | \(\text {Expression too large to display}\) | \(1269\) |
Input:
int(x^4*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RET URNVERBOSE)
Output:
-1/3*B*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f+1/3/b/d/f*(-1/f*(3*A*b*d*f +2*B*a*d*f-2*B*b*c*f-3*B*b*d*e)*c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c )^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(b/a)^(1/2),(- 1-(a*d-b*c)/c/b)^(1/2))-EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))) -(3*A*b*d*e*f-B*a*c*f^2-3*B*b*d*e^2)/f^2/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+ d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2 ),(-1-(a*d-b*c)/c/b)^(1/2))+3*b*d*e*(A*f-B*e)/f^2/(b/a)^(1/2)*(1-b*x^2/a)^ (1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticPi(x* (b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2)))*((-b*x^2+a)*(d*x^2+c))^( 1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^4*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="fricas")
Output:
Timed out
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{4} \left (A + B x^{2}\right )}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**4*(B*x**2+A)/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e), x)
Output:
Integral(x**4*(A + B*x**2)/(sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2) ), x)
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="maxima")
Output:
integrate((B*x^2 + A)*x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo rithm="giac")
Output:
integrate((B*x^2 + A)*x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((x^4*(A + B*x^2))/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int((x^4*(A + B*x^2))/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x )
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) a d f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b c f -3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b d e +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) a c f +2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) a d e -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b c e +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) a c e}{3 d f} \] Input:
int(x^4*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a - b*x**2)*x + 5*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*a*d*f - 2*int((sqrt(c + d*x**2) *sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b* c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*b*c*f - 3*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f* x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*b*d*e + int(( sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6),x)*a*c*f + 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d *e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d*f*x**6), x)*a*d*e - 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c*f*x **2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b*d *f*x**6),x)*b*c*e + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e + a*c*f *x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - b *d*f*x**6),x)*a*c*e)/(3*d*f)