Integrand size = 26, antiderivative size = 330 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {\left (7 a b-\frac {2 b^2 c}{d}+\frac {3 a^2 d}{c}\right ) x \sqrt {a+b x^2}}{3 \sqrt {c+d x^2}}+\frac {b (b c+3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {\left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} (b c-9 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
1/3*(7*a*b-2*b^2*c/d+3*a^2*d/c)*x*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*(-3* a^2*d^2-7*a*b*c*d+2*b^2*c^2)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellip ticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*(b*x^2+a)^(1/2 )/d^(3/2)/c^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*(- 9*a*d+b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^( 1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*(b*x^2+a)^(1/2)/d^(3/2)/ (c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-a*(b*x^2+a)^(3/2)*(d*x^2+c )^(1/2)/c/x+1/3*b*(3*a*d+b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/d
Result contains complex when optimal does not.
Time = 2.84 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (3 a^2 d-b^2 c x^2\right ) \left (c+d x^2\right )-i b c \left (-2 b^2 c^2+7 a b c d+3 a^2 d^2\right ) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i b c \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c d^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
(-(Sqrt[b/a]*d*(a + b*x^2)*(3*a^2*d - b^2*c*x^2)*(c + d*x^2)) - I*b*c*(-2* b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c] *EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*b*c*(b^2*c^2 - 4*a *b*c*d + 3*a^2*d^2)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I* ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c*d^2*x*Sqrt[a + b*x^2]*S qrt[c + d*x^2])
Time = 0.47 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {376, 27, 403, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 376 |
| \(\displaystyle \frac {\int \frac {b \sqrt {b x^2+a} \left ((b c+3 a d) x^2+4 a c\right )}{\sqrt {d x^2+c}}dx}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \left ((b c+3 a d) x^2+4 a c\right )}{\sqrt {d x^2+c}}dx}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {\int -\frac {\left (2 b^2 c^2-7 a b d c-3 a^2 d^2\right ) x^2+a c (b c-9 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}-\frac {\int \frac {\left (2 b^2 c^2-7 a b d c-3 a^2 d^2\right ) x^2+a c (b c-9 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}-\frac {\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-9 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}-\frac {\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}-\frac {\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 d}-\frac {\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}\right )}{c}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}\) |
-((a*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(c*x)) + (b*(((b*c + 3*a*d)*x*Sqrt [a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - ((2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)* ((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellipt icE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b*c - 9*a*d)*Sqrt[ a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[ d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d)))/c
3.10.62.3.1 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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 )/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^ 2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & & IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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 = 5.08 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.11
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {a^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{c x}+\frac {b^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 d}+\frac {\left (3 a^{2} b -\frac {a \,b^{2} c}{3 d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (3 a \,b^{2}+\frac {b d \,a^{2}}{c}-\frac {b^{2} \left (2 a d +2 b c \right )}{3 d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(365\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-b^{2} c \,x^{2}+3 a^{2} d \right )}{3 d c x}+\frac {b \left (\frac {9 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (3 a^{2} d^{2}+7 a b c d -2 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 c d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(422\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (\sqrt {-\frac {b}{a}}\, b^{3} c \,d^{2} x^{6}-3 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{4}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{4}+\sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{4}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x +7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x -3 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{2}-3 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{2}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{2}-3 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) d^{2} \sqrt {-\frac {b}{a}}\, c x}\) | \(568\) |
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-a^2/c*(b*d*x ^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+1/3*b^2/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^ (1/2)+(3*a^2*b-1/3*a*b^2*c/d)/(-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*a*b^2+b*d*a^2/c-1/3*b^2/d*(2*a*d+2*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))))
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{2}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{2} \sqrt {c + d x^{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{2}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^2\,\sqrt {d\,x^2+c}} \,d x \]