Integrand size = 28, antiderivative size = 258 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {e x}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+5 a d (2 b c-3 a d)\right ) \sqrt {e x}}{6 c^3 d e^3 \sqrt {c+d x^2}}+\frac {\left (b^2 c^2+5 a d (2 b c-3 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 c^{13/4} d^{5/4} e^{5/2} \sqrt {c+d x^2}} \]
-2/3*a^2/c/e/(e*x)^(3/2)/(d*x^2+c)^(3/2)-1/3*(3*a^2*d^2-2*a*b*c*d+b^2*c^2) *(e*x)^(1/2)/c^2/d/e^3/(d*x^2+c)^(3/2)+1/6*(b^2*c^2+5*a*d*(-3*a*d+2*b*c))* (e*x)^(1/2)/c^3/d/e^3/(d*x^2+c)^(1/2)+1/12*(b^2*c^2+5*a*d*(-3*a*d+2*b*c))* (cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan( d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(d^(1/4)*(e*x) ^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1 /2)+x*d^(1/2))^2)^(1/2)/c^(13/4)/d^(5/4)/e^(5/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {x^{5/2} \left (\frac {b^2 c^2 x^2 \left (-c+d x^2\right )+2 a b c d x^2 \left (7 c+5 d x^2\right )-a^2 d \left (4 c^2+21 c d x^2+15 d^2 x^4\right )}{c^3 d x^{3/2} \left (c+d x^2\right )}+\frac {i \left (b^2 c^2+10 a b c d-15 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{c^3 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d}\right )}{6 (e x)^{5/2} \sqrt {c+d x^2}} \]
(x^(5/2)*((b^2*c^2*x^2*(-c + d*x^2) + 2*a*b*c*d*x^2*(7*c + 5*d*x^2) - a^2* d*(4*c^2 + 21*c*d*x^2 + 15*d^2*x^4))/(c^3*d*x^(3/2)*(c + d*x^2)) + (I*(b^2 *c^2 + 10*a*b*c*d - 15*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[ Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^3*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d) ))/(6*(e*x)^(5/2)*Sqrt[c + d*x^2])
Time = 0.36 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {365, 27, 362, 253, 266, 761}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {3 \left (b^2 c x^2+a (2 b c-3 a d)\right )}{2 \sqrt {e x} \left (d x^2+c\right )^{5/2}}dx}{3 c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^2 c x^2+a (2 b c-3 a d)}{\sqrt {e x} \left (d x^2+c\right )^{5/2}}dx}{c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a (2 b c-3 a d)}{c}+\frac {b^2 c}{d}\right ) \int \frac {1}{\sqrt {e x} \left (d x^2+c\right )^{3/2}}dx-\frac {\sqrt {e x} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c d e \left (c+d x^2\right )^{3/2}}}{c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a (2 b c-3 a d)}{c}+\frac {b^2 c}{d}\right ) \left (\frac {\int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx}{2 c}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )-\frac {\sqrt {e x} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c d e \left (c+d x^2\right )^{3/2}}}{c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a (2 b c-3 a d)}{c}+\frac {b^2 c}{d}\right ) \left (\frac {\int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{c e}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )-\frac {\sqrt {e x} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c d e \left (c+d x^2\right )^{3/2}}}{c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a (2 b c-3 a d)}{c}+\frac {b^2 c}{d}\right ) \left (\frac {\left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 c^{5/4} \sqrt [4]{d} e^{3/2} \sqrt {c+d x^2}}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )-\frac {\sqrt {e x} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c d e \left (c+d x^2\right )^{3/2}}}{c e^2}-\frac {2 a^2}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}\) |
(-2*a^2)/(3*c*e*(e*x)^(3/2)*(c + d*x^2)^(3/2)) + (-1/3*((b^2*c^2 - 2*a*b*c *d + 3*a^2*d^2)*Sqrt[e*x])/(c*d*e*(c + d*x^2)^(3/2)) + (((b^2*c)/d + (5*a* (2*b*c - 3*a*d))/c)*(Sqrt[e*x]/(c*e*Sqrt[c + d*x^2]) + ((Sqrt[c]*e + Sqrt[ d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[c]*e + Sqrt[d]*e*x)^2]*EllipticF[2* ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(2*c^(5/4)*d^(1/4)*e^ (3/2)*Sqrt[c + d*x^2])))/6)/(c*e^2)
3.9.63.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 4.82 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.23
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 c^{2} e^{3} d^{3} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {x \left (11 a^{2} d^{2}-10 a b c d -b^{2} c^{2}\right )}{6 d \,e^{2} c^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{3 c^{3} e^{3} x^{2}}+\frac {\left (-\frac {11 a^{2} d^{2}-10 a b c d -b^{2} c^{2}}{12 d \,c^{3} e^{2}}-\frac {d \,a^{2}}{3 c^{3} e^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(318\) |
| risch | \(-\frac {2 a^{2} \sqrt {d \,x^{2}+c}}{3 c^{3} x \,e^{2} \sqrt {e x}}-\frac {\left (\frac {a^{2} \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\frac {3 c \left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )}{d}+\frac {3 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {5 x}{6 c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {5 \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{12 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{3 c^{3} e^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(529\) |
| default | \(-\frac {15 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{3} x^{3}-10 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c \,d^{2} x^{3}-\sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} d \,x^{3}+15 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x -10 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d x -\sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x +30 a^{2} d^{4} x^{4}-20 c a b \,x^{4} d^{3}-2 b^{2} c^{2} d^{2} x^{4}+42 a^{2} c \,d^{3} x^{2}-28 a b \,c^{2} d^{2} x^{2}+2 b^{2} c^{3} d \,x^{2}+8 a^{2} c^{2} d^{2}}{12 x \,e^{2} \sqrt {e x}\, c^{3} d^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(686\) |
(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-1/3/c^2/e^3/d^3*(a^2*d ^2-2*a*b*c*d+b^2*c^2)*(d*e*x^3+c*e*x)^(1/2)/(x^2+c/d)^2-1/6/d/e^2*x/c^3*(1 1*a^2*d^2-10*a*b*c*d-b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)-2/3/c^3/e^3*a^2*(d*e *x^3+c*e*x)^(1/2)/x^2+(-1/12/d/c^3*(11*a^2*d^2-10*a*b*c*d-b^2*c^2)/e^2-1/3 *d/c^3/e^2*a^2)*(-c*d)^(1/2)/d*((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*( -2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e *x^3+c*e*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2* 2^(1/2)))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (b^{2} c^{2} d^{2} + 10 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{6} + 2 \, {\left (b^{2} c^{3} d + 10 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{4} + {\left (b^{2} c^{4} + 10 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (4 \, a^{2} c^{2} d^{2} - {\left (b^{2} c^{2} d^{2} + 10 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{4} + {\left (b^{2} c^{3} d - 14 \, a b c^{2} d^{2} + 21 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{6 \, {\left (c^{3} d^{4} e^{3} x^{6} + 2 \, c^{4} d^{3} e^{3} x^{4} + c^{5} d^{2} e^{3} x^{2}\right )}} \]
1/6*(((b^2*c^2*d^2 + 10*a*b*c*d^3 - 15*a^2*d^4)*x^6 + 2*(b^2*c^3*d + 10*a* b*c^2*d^2 - 15*a^2*c*d^3)*x^4 + (b^2*c^4 + 10*a*b*c^3*d - 15*a^2*c^2*d^2)* x^2)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) - (4*a^2*c^2*d^2 - (b^2*c ^2*d^2 + 10*a*b*c*d^3 - 15*a^2*d^4)*x^4 + (b^2*c^3*d - 14*a*b*c^2*d^2 + 21 *a^2*c*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(c^3*d^4*e^3*x^6 + 2*c^4*d^3*e ^3*x^4 + c^5*d^2*e^3*x^2)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]