Integrand size = 26, antiderivative size = 378 \[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {4 d x \left (a+b x^2\right )}{3 b^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d (7 b c-8 a d) x \left (a+b x^2\right )}{3 b^3 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {x \left (c+d x^2\right )}{b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} (7 b c-8 a d) \left (a+b x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^3 e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {c^{3/2} (3 b c-4 a d) \left (a+b x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b^2 \sqrt {d} e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \]
4/3*d*x*(b*x^2+a)/b^2/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/3*d*(-8*a*d+7*b*c) *x*(b*x^2+a)/b^3/e/(d*x^2+c)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-x*(d*x^2+c)/b/e /(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/3*c^(3/2)*(-4*a*d+3*b*c)*(b*x^2+a)*(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))/a/b^2/e/(d*x^2+c)/d^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c ))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/3*(-8*a*d+7*b*c)*(b*x^2+a)*(1/(1+ d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^ (1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*d^(1/2)/b^3/e/(d*x^2+c)/(c*(b*x^2+a)/a/(d *x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 4.91 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.58 \[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (-3 b c+4 a d+b d x^2\right )+i c (-7 b c+8 a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-4 i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \left (\frac {b}{a}\right )^{5/2} e^2 \left (a+b x^2\right )} \]
(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*x*(c + d*x^2)*(-3*b*c + 4*a* d + b*d*x^2) + I*c*(-7*b*c + 8*a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c ]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (4*I)*c*(-(b*c) + a*d)* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^2*(b/a)^(5/2)*e^2*(a + b*x^2))
Time = 0.48 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2058, 369, 403, 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.
| \(\displaystyle \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \int \frac {x^2 \left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right )^{3/2}}dx}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\int \frac {\sqrt {d x^2+c} \left (4 d x^2+c\right )}{\sqrt {b x^2+a}}dx}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\frac {\int \frac {d (7 b c-8 a d) x^2+c (3 b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {4 d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\frac {c (3 b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+d (7 b c-8 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {4 d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\frac {d (7 b c-8 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {4 d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\frac {d (7 b c-8 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b}+\frac {4 d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d (7 b c-8 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b}+\frac {4 d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}}{b}-\frac {x \left (c+d x^2\right )^{3/2}}{b \sqrt {a+b x^2}}\right )}{e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
(Sqrt[a + b*x^2]*(-((x*(c + d*x^2)^(3/2))/(b*Sqrt[a + b*x^2])) + ((4*d*x*S qrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b) + (d*(7*b*c - 8*a*d)*((x*Sqrt[a + b* x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqr t[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)*(3*b*c - 4*a*d)*Sqrt[a + b*x^2]*El lipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c* (a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*b))/b))/(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
3.4.14.3.1 Defintions of rubi rules used
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[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[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]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 8.86 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(\frac {\left (b \,x^{2}+a \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b c d \,x^{3}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b c d \,x^{3}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d -4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d +7 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a c d x +3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a c d x -3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b \,c^{2} x \right )}{3 b^{2} {\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(643\) |
| risch | \(\frac {d x \left (b \,x^{2}+a \right )}{3 b^{2} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}-\frac {\left (\frac {3 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\left (b d e \,x^{2}+e b c \right ) x}{a \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d e \,x^{2}+e b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{a \left (a d -b c \right )}\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 {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d b c e \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 {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right )}{b}+\frac {-\frac {3 a^{2} d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {3 b^{2} c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {7 a b 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 {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 \left (5 a b \,d^{2}-4 b^{2} c d \right ) a c e \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 {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}}{b}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{3 b^{2} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(942\) |
1/3*(b*x^2+a)*(((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*b*d^2*x^5+((d*x^2+ c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*a*d^2*x^3+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b /a)^(1/2)*b*c*d*x^3+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*d ^2*x^3-3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*b*c*d*x^3+4*((d* x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF( x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*c*d-4*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^ 2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2) )*b*c^2-8*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1 /2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*c*d+7*((d*x^2+c)*(b*x^2+a) )^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),( a*d/b/c)^(1/2))*b*c^2+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*a*c*d*x+3*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*c*d*x-3*(b*d*x^4+a*d*x^2 +b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*b*c^2*x)/b^2/(e*(b*x^2+a)/(d*x^2+c))^(3/2 )/(d*x^2+c)^2/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
Time = 0.12 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=-\frac {{\left ({\left (7 \, b^{2} c^{2} - 8 \, a b c d\right )} x^{3} + {\left (7 \, a b c^{2} - 8 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b e}{d}} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (7 \, b^{2} c^{2} - 4 \, a b d^{2} - {\left (8 \, a b - 3 \, b^{2}\right )} c d\right )} x^{3} + {\left (7 \, a b c^{2} - 4 \, a^{2} d^{2} - {\left (8 \, a^{2} - 3 \, a b\right )} c d\right )} x\right )} \sqrt {\frac {b e}{d}} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} d^{2} x^{6} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{4} + 7 \, a b c^{2} - 8 \, a^{2} c d + {\left (4 \, b^{2} c^{2} + 3 \, a b c d - 8 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, {\left (b^{4} e^{2} x^{3} + a b^{3} e^{2} x\right )}} \]
-1/3*(((7*b^2*c^2 - 8*a*b*c*d)*x^3 + (7*a*b*c^2 - 8*a^2*c*d)*x)*sqrt(b*e/d )*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((7*b^2*c^2 - 4 *a*b*d^2 - (8*a*b - 3*b^2)*c*d)*x^3 + (7*a*b*c^2 - 4*a^2*d^2 - (8*a^2 - 3* a*b)*c*d)*x)*sqrt(b*e/d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/( b*c)) - (b^2*d^2*x^6 + (5*b^2*c*d - 4*a*b*d^2)*x^4 + 7*a*b*c^2 - 8*a^2*c*d + (4*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^4*e^2*x^3 + a*b^3*e^2*x)
Timed out. \[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {x^{2}}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {x^{2}}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]