Integrand size = 30, antiderivative size = 373 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {f (b c (d e-8 c f)+2 a d (d e+c f)) x \sqrt {c+d x^2}}{3 c^2 d^3 \sqrt {e+f x^2}}+\frac {(b c (d e-4 c f)+a d (2 d e+c f)) x \sqrt {e+f x^2}}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (e+f x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \sqrt {f} (b c (d e-8 c f)+2 a d (d e+c f)) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 d^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(4 b c-a d) e^{3/2} \sqrt {f} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 c^2 d^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
-1/3*(-a*d+b*c)*x*(f*x^2+e)^(3/2)/c/d/(d*x^2+c)^(3/2)-1/3*f*(b*c*(-8*c*f+d *e)+2*a*d*(c*f+d*e))*x*(d*x^2+c)^(1/2)/c^2/d^3/(f*x^2+e)^(1/2)+1/3*(-a*d+4 *b*c)*e^(3/2)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/ e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*f^(1/2)*(d*x^2+c)^(1/2)/c^2/d ^2/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*(b*c*(-8*c*f+d*e)+2 *a*d*(c*f+d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2 )/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*f^(1/2)*(d*x^2+c)^( 1/2)/c^2/d^3/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*(b*c*(-4* c*f+d*e)+a*d*(c*f+2*d*e))*x*(f*x^2+e)^(1/2)/c^2/d^2/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 5.58 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (\frac {d}{c}\right )^{3/2} \left (\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (b c \left (-4 c^2 f+d^2 e x^2-5 c d f x^2\right )+a d \left (c^2 f+2 d^2 e x^2+c d \left (3 e+2 f x^2\right )\right )\right )-i e (-2 a d (d e+c f)+b c (-d e+8 c f)) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i e (-a d (2 d e+c f)+b c (-d e+4 c f)) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{3 d^4 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \]
((d/c)^(3/2)*(Sqrt[d/c]*x*(e + f*x^2)*(b*c*(-4*c^2*f + d^2*e*x^2 - 5*c*d*f *x^2) + a*d*(c^2*f + 2*d^2*e*x^2 + c*d*(3*e + 2*f*x^2))) - I*e*(-2*a*d*(d* e + c*f) + b*c*(-(d*e) + 8*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*e*(-(a*d*(2* d*e + c*f)) + b*c*(-(d*e) + 4*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(3*d^4*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])
Time = 0.52 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {401, 25, 401, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {f x^2+e} \left ((4 b c-a d) f x^2+(b c+2 a d) e\right )}{\left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {f x^2+e} \left ((4 b c-a d) f x^2+(b c+2 a d) e\right )}{\left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}-\frac {\int -\frac {f \left (c (4 b c-a d) e-(b c (d e-8 c f)+2 a d (d e+c f)) x^2\right )}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{c d}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {f \left (c (4 b c-a d) e-(b c (d e-8 c f)+2 a d (d e+c f)) x^2\right )}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {f \int \frac {c (4 b c-a d) e-(b c (d e-8 c f)+2 a d (d e+c f)) x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {f \left (c e (4 b c-a d) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx-(2 a d (c f+d e)+b c (d e-8 c f)) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx\right )}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {f \left (\frac {e^{3/2} \sqrt {c+d x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-(2 a d (c f+d e)+b c (d e-8 c f)) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx\right )}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {f \left (\frac {e^{3/2} \sqrt {c+d x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-(2 a d (c f+d e)+b c (d e-8 c f)) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )\right )}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {f \left (\frac {e^{3/2} \sqrt {c+d x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-(2 a d (c f+d e)+b c (d e-8 c f)) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )\right )}{c d}+\frac {x \sqrt {e+f x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}}{3 c d}-\frac {x \left (e+f x^2\right )^{3/2} (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}\) |
-1/3*((b*c - a*d)*x*(e + f*x^2)^(3/2))/(c*d*(c + d*x^2)^(3/2)) + (((b*c*(d *e - 4*c*f) + a*d*(2*d*e + c*f))*x*Sqrt[e + f*x^2])/(c*d*Sqrt[c + d*x^2]) + (f*(-((b*c*(d*e - 8*c*f) + 2*a*d*(d*e + c*f))*((x*Sqrt[c + d*x^2])/(d*Sq rt[e + f*x^2]) - (Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqr t[e]], 1 - (d*e)/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]* Sqrt[e + f*x^2]))) + ((4*b*c - a*d)*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcT an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[f]*Sqrt[(e*(c + d*x^2))/( c*(e + f*x^2))]*Sqrt[e + f*x^2])))/(c*d))/(3*c*d)
3.1.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Time = 4.50 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.50
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {\left (a c d f -a e \,d^{2}-c^{2} b f +b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) \left (2 a c d f +2 a e \,d^{2}-5 c^{2} b f +b c d e \right ) x}{3 c^{2} d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (a d f -2 b c f +2 b d e \right )}{d^{3}}-\frac {\left (a c d f -a e \,d^{2}-c^{2} b f +b c d e \right ) f}{3 d^{3} c}-\frac {\left (2 a c d f +2 a e \,d^{2}-5 c^{2} b f +b c d e \right ) \left (c f -d e \right )}{3 d^{3} c^{2}}-\frac {e \left (2 a c d f +2 a e \,d^{2}-5 c^{2} b f +b c d e \right )}{3 d^{2} c^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (\frac {b \,f^{2}}{d^{2}}-\frac {\left (2 a c d f +2 a e \,d^{2}-5 c^{2} b f +b c d e \right ) f}{3 d^{2} c^{2}}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(559\) |
| default | \(\text {Expression too large to display}\) | \(1225\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(-1/3*(a*c*d*f -a*d^2*e-b*c^2*f+b*c*d*e)/c/d^4*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(x^2 +c/d)^2+1/3*(d*f*x^2+d*e)*(2*a*c*d*f+2*a*d^2*e-5*b*c^2*f+b*c*d*e)/c^2/d^3* x/((x^2+c/d)*(d*f*x^2+d*e))^(1/2)+(f*(a*d*f-2*b*c*f+2*b*d*e)/d^3-1/3*(a*c* d*f-a*d^2*e-b*c^2*f+b*c*d*e)/d^3*f/c-1/3*(2*a*c*d*f+2*a*d^2*e-5*b*c^2*f+b* c*d*e)/d^3*(c*f-d*e)/c^2-1/3/d^2*e*(2*a*c*d*f+2*a*d^2*e-5*b*c^2*f+b*c*d*e) /c^2)/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d* e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-(b*f^2 /d^2-1/3*(2*a*c*d*f+2*a*d^2*e-5*b*c^2*f+b*c*d*e)/d^2*f/c^2)*e/(-d/c)^(1/2) *(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/f *(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-EllipticE(x*(-d/c)^(1 /2),(-1+(c*f+d*e)/e/d)^(1/2))))
Time = 0.11 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (b c d^{3} + 2 \, a d^{4}\right )} e^{2} - 2 \, {\left (4 \, b c^{2} d^{2} - a c d^{3}\right )} e f\right )} x^{5} + 2 \, {\left ({\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} e^{2} - 2 \, {\left (4 \, b c^{3} d - a c^{2} d^{2}\right )} e f\right )} x^{3} + {\left ({\left (b c^{3} d + 2 \, a c^{2} d^{2}\right )} e^{2} - 2 \, {\left (4 \, b c^{4} - a c^{3} d\right )} e f\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left ({\left ({\left (b c d^{3} + 2 \, a d^{4}\right )} e^{2} - 2 \, {\left (4 \, b c^{2} d^{2} - a c d^{3}\right )} e f - {\left (4 \, b c^{2} d^{2} - a c d^{3}\right )} f^{2}\right )} x^{5} + 2 \, {\left ({\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} e^{2} - 2 \, {\left (4 \, b c^{3} d - a c^{2} d^{2}\right )} e f - {\left (4 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{3} + {\left ({\left (b c^{3} d + 2 \, a c^{2} d^{2}\right )} e^{2} - 2 \, {\left (4 \, b c^{4} - a c^{3} d\right )} e f - {\left (4 \, b c^{4} - a c^{3} d\right )} f^{2}\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) + {\left (3 \, b c^{2} d^{2} f^{2} x^{4} - {\left (b c^{3} d + 2 \, a c^{2} d^{2}\right )} e f + 2 \, {\left (4 \, b c^{4} - a c^{3} d\right )} f^{2} - {\left ({\left (2 \, b c^{2} d^{2} + a c d^{3}\right )} e f - 3 \, {\left (4 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{3 \, {\left (c^{2} d^{5} f x^{5} + 2 \, c^{3} d^{4} f x^{3} + c^{4} d^{3} f x\right )}} \]
1/3*((((b*c*d^3 + 2*a*d^4)*e^2 - 2*(4*b*c^2*d^2 - a*c*d^3)*e*f)*x^5 + 2*(( b*c^2*d^2 + 2*a*c*d^3)*e^2 - 2*(4*b*c^3*d - a*c^2*d^2)*e*f)*x^3 + ((b*c^3* d + 2*a*c^2*d^2)*e^2 - 2*(4*b*c^4 - a*c^3*d)*e*f)*x)*sqrt(d*f)*sqrt(-e/f)* elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (((b*c*d^3 + 2*a*d^4)*e^2 - 2*(4*b*c^2*d^2 - a*c*d^3)*e*f - (4*b*c^2*d^2 - a*c*d^3)*f^2)*x^5 + 2*((b*c ^2*d^2 + 2*a*c*d^3)*e^2 - 2*(4*b*c^3*d - a*c^2*d^2)*e*f - (4*b*c^3*d - a*c ^2*d^2)*f^2)*x^3 + ((b*c^3*d + 2*a*c^2*d^2)*e^2 - 2*(4*b*c^4 - a*c^3*d)*e* f - (4*b*c^4 - a*c^3*d)*f^2)*x)*sqrt(d*f)*sqrt(-e/f)*elliptic_f(arcsin(sqr t(-e/f)/x), c*f/(d*e)) + (3*b*c^2*d^2*f^2*x^4 - (b*c^3*d + 2*a*c^2*d^2)*e* f + 2*(4*b*c^4 - a*c^3*d)*f^2 - ((2*b*c^2*d^2 + a*c*d^3)*e*f - 3*(4*b*c^3* d - a*c^2*d^2)*f^2)*x^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(c^2*d^5*f*x^5 + 2*c^3*d^4*f*x^3 + c^4*d^3*f*x)
\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]