Integrand size = 32, antiderivative size = 279 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {f^2 x \sqrt {a+b x^2}}{b d \sqrt {c+d x^2}}+\frac {\left (2 b c f (d e-c f)-d \left (b d e^2-a c f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {c} d^{3/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (b d e^2-a f (2 d e-c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a d^{3/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
f^2*x*(b*x^2+a)^(1/2)/b/d/(d*x^2+c)^(1/2)+(2*b*c*f*(-c*f+d*e)-d*(-a*c*f^2+ b*d*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1 -b*c/a/d)^(1/2))/b/c^(1/2)/d^(3/2)/(-a*d+b*c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1 /2)/(d*x^2+c)^(1/2)+c^(1/2)*(b*d*e^2-a*f*(-c*f+2*d*e))*(b*x^2+a)^(1/2)*Inv erseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/d^(3/2)/(-a*d+ b*c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 10.70 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.88 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {-i c \left (a c d f^2-b \left (d^2 e^2-2 c d e f+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(d e-c f) \left (\sqrt {\frac {b}{a}} d (d e-c f) x \left (a+b x^2\right )-2 i c (-b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c d^2 (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]
Output:
((-I)*c*(a*c*d*f^2 - b*(d^2*e^2 - 2*c*d*e*f + 2*c^2*f^2))*Sqrt[1 + (b*x^2) /a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + ( d*e - c*f)*(Sqrt[b/a]*d*(d*e - c*f)*x*(a + b*x^2) - (2*I)*c*(-(b*c) + a*d) *f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x ], (a*d)/(b*c)]))/(Sqrt[b/a]*c*d^2*(-(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. \(695\) vs. \(2(279)=558\).
Time = 0.90 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.49, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {2 e f x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {f^2 x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{d^{3/2} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {c} f^2 \sqrt {a+b x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b d^{3/2} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b \sqrt {c} e^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} e^2 \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 \sqrt {c} e f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 \sqrt {c} e f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {f^2 x \sqrt {a+b x^2} (2 b c-a d)}{b d \sqrt {c+d x^2} (b c-a d)}-\frac {c f^2 x \sqrt {a+b x^2}}{d \sqrt {c+d x^2} (b c-a d)}\) |
Input:
Int[(e + f*x^2)^2/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)),x]
Output:
-((c*f^2*x*Sqrt[a + b*x^2])/(d*(b*c - a*d)*Sqrt[c + d*x^2])) + ((2*b*c - a *d)*f^2*x*Sqrt[a + b*x^2])/(b*d*(b*c - a*d)*Sqrt[c + d*x^2]) - (Sqrt[d]*e^ 2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)]) /(Sqrt[c]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 ]) + (2*Sqrt[c]*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 ))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(2*b*c - a*d)*f^2*Sqrt[a + b*x^2]*Elliptic E[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*d^(3/2)*(b*c - a*d)*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*e^2*Sqrt [a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sq rt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt[c]*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]* Sqrt[c + d*x^2]) + (c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]* x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(d^(3/2)*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/( a*(c + d*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 = 6.30 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.61
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) x \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{c \,d^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (-\frac {f \left (c f -2 d e \right )}{d^{2}}+\frac {c^{2} f^{2}-2 c d e f +d^{2} e^{2}}{d^{2} c}-\frac {a \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{d c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {f^{2}}{d}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b}{d \left (a d -b c \right ) c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(449\) |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, b \,c^{2} d \,f^{2} x^{3}-2 \sqrt {-\frac {b}{a}}\, b c \,d^{2} e f \,x^{3}+\sqrt {-\frac {b}{a}}\, b \,d^{3} e^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,c^{2} d \,f^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c \,d^{2} e f +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{3} f^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} d e f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,c^{2} d \,f^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{3} f^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} d e f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c \,d^{2} e^{2}+\sqrt {-\frac {b}{a}}\, a \,c^{2} d \,f^{2} x -2 \sqrt {-\frac {b}{a}}\, a c \,d^{2} e f x +\sqrt {-\frac {b}{a}}\, a \,d^{3} e^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{c \,d^{2} \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(639\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),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+a*d) /c/d^2/(a*d-b*c)*x*(c^2*f^2-2*c*d*e*f+d^2*e^2)/((x^2+c/d)*(b*d*x^2+a*d))^( 1/2)+(-f*(c*f-2*d*e)/d^2+(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^2/c-a/d/c/(a*d-b*c) *(c^2*f^2-2*c*d*e*f+d^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/d-(c^2*f^2-2*c*d*e*f+d^2*e^2)/d/(a*d-b*c)*b/c)*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))))
Time = 0.10 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.55 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (b c d^{3} e^{2} - 2 \, b c^{2} d^{2} e f + {\left (2 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{3} + {\left (b c^{2} d^{2} e^{2} - 2 \, b c^{3} d e f + {\left (2 \, b c^{4} - a c^{3} d\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left ({\left (b c d^{3} + b d^{4}\right )} e^{2} - 2 \, {\left (b c^{2} d^{2} + a d^{4}\right )} e f + {\left (2 \, b c^{3} d - a c^{2} d^{2} + a c d^{3}\right )} f^{2}\right )} x^{3} + {\left ({\left (b c^{2} d^{2} + b c d^{3}\right )} e^{2} - 2 \, {\left (b c^{3} d + a c d^{3}\right )} e f + {\left (2 \, b c^{4} - a c^{3} d + a c^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b c d^{3} e^{2} - 2 \, b c^{2} d^{2} e f + {\left (b c^{2} d^{2} - a c d^{3}\right )} f^{2} x^{2} + {\left (2 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (b^{2} c^{2} d^{4} - a b c d^{5}\right )} x^{3} + {\left (b^{2} c^{3} d^{3} - a b c^{2} d^{4}\right )} x} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="fricas ")
Output:
-(((b*c*d^3*e^2 - 2*b*c^2*d^2*e*f + (2*b*c^3*d - a*c^2*d^2)*f^2)*x^3 + (b* c^2*d^2*e^2 - 2*b*c^3*d*e*f + (2*b*c^4 - a*c^3*d)*f^2)*x)*sqrt(b*d)*sqrt(- c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (((b*c*d^3 + b*d^4)*e^2 - 2*(b*c^2*d^2 + a*d^4)*e*f + (2*b*c^3*d - a*c^2*d^2 + a*c*d^3)*f^2)*x^3 + ((b*c^2*d^2 + b*c*d^3)*e^2 - 2*(b*c^3*d + a*c*d^3)*e*f + (2*b*c^4 - a*c^ 3*d + a*c^2*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d) /x), a*d/(b*c)) - (b*c*d^3*e^2 - 2*b*c^2*d^2*e*f + (b*c^2*d^2 - a*c*d^3)*f ^2*x^2 + (2*b*c^3*d - a*c^2*d^2)*f^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/((b ^2*c^2*d^4 - a*b*c*d^5)*x^3 + (b^2*c^3*d^3 - a*b*c^2*d^4)*x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2),x)
Output:
Integral((e + f*x**2)**2/(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, e f x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,c^{2} f^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d e f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d \,f^{2} x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,d^{2} e f \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a \,c^{2} e f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a c d e f \,x^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b \,c^{2} e^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b c d \,e^{2} x^{2}}{b c \left (d \,x^{2}+c \right )} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),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*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d *x**4 + b*d**2*x**6),x)*b*c**2*f**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x** 2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b*c*d*e*f + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4 )/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d* *2*x**6),x)*b*c*d*f**2*x**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) /(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d** 2*x**6),x)*b*d**2*e*f*x**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c* *2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6 ),x)*a*c**2*e*f - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c* d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*c*d* e*f*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b*c**2*e**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x **4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b*c*d*e**2*x**2)/(b*c*( c + d*x**2))