Integrand size = 33, antiderivative size = 283 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {(b e+a f)^2 x \sqrt {c+d x^2}}{a b (b c+a d) \sqrt {a-b x^2}}-\frac {\left (b^2 d e^2+2 a^2 d f^2+a b f (2 d e+c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} b^{3/2} d (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\left (b d e^2+a c f^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
(a*f+b*e)^2*x*(d*x^2+c)^(1/2)/a/b/(a*d+b*c)/(-b*x^2+a)^(1/2)-(b^2*d*e^2+2* a^2*d*f^2+a*b*f*(c*f+2*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))/a^(1/2)/b^(3/2)/d/(a*d+b*c)/(-b*x^2+a)^ (1/2)/(1+d*x^2/c)^(1/2)+(a*c*f^2+b*d*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))/a^(1/2)/b^(3/2)/d/(-b*x^ 2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 9.40 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.89 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {b \left (\sqrt {-\frac {b}{a}} d (b e+a f)^2 x \left (c+d x^2\right )+i c \left (b^2 d e^2+2 a^2 d f^2+a b f (2 d e+c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i (b c+a d) \left (b d e^2+a c f^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{a^3 \left (-\frac {b}{a}\right )^{5/2} d (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(b*(Sqrt[-(b/a)]*d*(b*e + a*f)^2*x*(c + d*x^2) + I*c*(b^2*d*e^2 + 2*a^2*d* f^2 + a*b*f*(2*d*e + c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti cE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*(b*c + a*d)*(b*d*e^2 + a *c*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))]))/(a^3*(-(b/a))^(5/2)*d*(b*c + a*d)*Sqrt[a - b* x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(283)=566\).
Time = 0.83 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {433, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 e f x^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {f^2 x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a} c f^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{b^{3/2} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} f^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (2 a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{b^{3/2} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c)}+\frac {e^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {b} e^2 \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c)}-\frac {2 \sqrt {a} e f \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1} (a d+b c)}+\frac {b e^2 x \sqrt {c+d x^2}}{a \sqrt {a-b x^2} (a d+b c)}+\frac {2 e f x \sqrt {c+d x^2}}{\sqrt {a-b x^2} (a d+b c)}+\frac {a f^2 x \sqrt {c+d x^2}}{b \sqrt {a-b x^2} (a d+b c)}\) |
Input:
Int[(e + f*x^2)^2/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(b*e^2*x*Sqrt[c + d*x^2])/(a*(b*c + a*d)*Sqrt[a - b*x^2]) + (2*e*f*x*Sqrt[ c + d*x^2])/((b*c + a*d)*Sqrt[a - b*x^2]) + (a*f^2*x*Sqrt[c + d*x^2])/(b*( b*c + a*d)*Sqrt[a - b*x^2]) - (Sqrt[b]*e^2*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d* x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[a]*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (2*Sqrt[a]*e*f*Sqrt[1 - (b* x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b* c))])/(Sqrt[b]*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a] *(b*c + 2*a*d)*f^2*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(S qrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(b^(3/2)*d*(b*c + a*d)*Sqrt[a - b*x^2 ]*Sqrt[1 + (d*x^2)/c]) + (e^2*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli pticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[a]*Sqrt[b]*Sqrt[ a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[a]*c*f^2*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(b^(3/2 )*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) ^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
Time = 6.21 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.63
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) x \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b^{2} a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {f \left (a f +2 b e \right )}{b^{2}}+\frac {a^{2} f^{2}+2 a b f e +b^{2} e^{2}}{b^{2} a}-\frac {c \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b a \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (-\frac {f^{2}}{b}-\frac {d \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b \left (a d +b c \right ) a}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(462\) |
| default | \(\frac {\left (\sqrt {\frac {b}{a}}\, a^{2} d^{2} f^{2} x^{3}+2 \sqrt {\frac {b}{a}}\, a b \,d^{2} e f \,x^{3}+\sqrt {\frac {b}{a}}\, b^{2} d^{2} e^{2} x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,d^{2} e^{2}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}-2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}-\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}-2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b c d e f -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}+\sqrt {\frac {b}{a}}\, a^{2} c d \,f^{2} x +2 \sqrt {\frac {b}{a}}\, a b c d e f x +\sqrt {\frac {b}{a}}\, b^{2} c d \,e^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{b a d \sqrt {\frac {b}{a}}\, \left (a d +b c \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(648\) |
Input:
int((f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(-b*d*x^2- b*c)/b^2/a/(a*d+b*c)*x*(a^2*f^2+2*a*b*e*f+b^2*e^2)/((x^2-a/b)*(-b*d*x^2-b* c))^(1/2)+(-1/b^2*f*(a*f+2*b*e)+(a^2*f^2+2*a*b*e*f+b^2*e^2)/b^2/a-1/b*c/a/ (a*d+b*c)*(a^2*f^2+2*a*b*e*f+b^2*e^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))-(-f^2/b-d/b*(a^2*f^2+2*a*b*e*f+b^2*e^2)/(a*d+b*c )/a)*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))-Ell ipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.52 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left ({\left (a b^{3} d e^{2} + 2 \, a^{2} b^{2} d e f + {\left (a^{2} b^{2} c + 2 \, a^{3} b d\right )} f^{2}\right )} x^{3} - {\left (a^{2} b^{2} d e^{2} + 2 \, a^{3} b d e f + {\left (a^{3} b c + 2 \, a^{4} d\right )} f^{2}\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left ({\left ({\left (a b^{3} - b^{4}\right )} d e^{2} + 2 \, {\left (b^{4} c + a^{2} b^{2} d\right )} e f + {\left (2 \, a^{3} b d + {\left (a^{2} b^{2} + a b^{3}\right )} c\right )} f^{2}\right )} x^{3} - {\left ({\left (a^{2} b^{2} - a b^{3}\right )} d e^{2} + 2 \, {\left (a b^{3} c + a^{3} b d\right )} e f + {\left (2 \, a^{4} d + {\left (a^{3} b + a^{2} b^{2}\right )} c\right )} f^{2}\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left (a b^{3} d e^{2} + 2 \, a^{2} b^{2} d e f - {\left (a b^{3} c + a^{2} b^{2} d\right )} f^{2} x^{2} + {\left (a^{2} b^{2} c + 2 \, a^{3} b d\right )} f^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} - {\left (a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x} \] Input:
integrate((f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="frica s")
Output:
(((a*b^3*d*e^2 + 2*a^2*b^2*d*e*f + (a^2*b^2*c + 2*a^3*b*d)*f^2)*x^3 - (a^2 *b^2*d*e^2 + 2*a^3*b*d*e*f + (a^3*b*c + 2*a^4*d)*f^2)*x)*sqrt(-b*d)*sqrt(a /b)*elliptic_e(arcsin(sqrt(a/b)/x), -b*c/(a*d)) - (((a*b^3 - b^4)*d*e^2 + 2*(b^4*c + a^2*b^2*d)*e*f + (2*a^3*b*d + (a^2*b^2 + a*b^3)*c)*f^2)*x^3 - ( (a^2*b^2 - a*b^3)*d*e^2 + 2*(a*b^3*c + a^3*b*d)*e*f + (2*a^4*d + (a^3*b + a^2*b^2)*c)*f^2)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), - b*c/(a*d)) - (a*b^3*d*e^2 + 2*a^2*b^2*d*e*f - (a*b^3*c + a^2*b^2*d)*f^2*x^ 2 + (a^2*b^2*c + 2*a^3*b*d)*f^2)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c))/((a*b^5 *c*d + a^2*b^4*d^2)*x^3 - (a^2*b^4*c*d + a^3*b^3*d^2)*x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((e + f*x**2)**2/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxim a")
Output:
integrate((f*x^2 + e)^2/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac" )
Output:
integrate((f*x^2 + e)^2/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((e + f*x^2)^2/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int((e + f*x^2)^2/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, e f x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d \,f^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d e f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d \,f^{2} x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d e f \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c e f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d \,e^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b c e f \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d \,x^{6}-2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d \,e^{2} x^{2}}{a d \left (-b \,x^{2}+a \right )} \] Input:
int((f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a - b*x**2)*e*f*x + int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c *x**4 + b**2*d*x**6),x)*a**2*d*f**2 + int((sqrt(c + d*x**2)*sqrt(a - b*x** 2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*d*e*f - int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4 )/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2 *d*x**6),x)*a*b*d*f**2*x**2 - int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4) /(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2* d*x**6),x)*b**2*d*e*f*x**2 - int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2 *c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6 ),x)*a**2*c*e*f + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c + a**2*d *x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*d *e**2 + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c + a**2*d*x**2 - 2* a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*c*e*f*x**2 - int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c + a**2*d*x**2 - 2*a*b*c*x **2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*d*e**2*x**2)/(a*d*( a - b*x**2))