Integrand size = 21, antiderivative size = 409 \[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {x \left (c+d x^2\right )}{a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {4 x \left (b+a c+a d x^2\right )}{3 a^2 d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(8 b+a c) x \left (b+a c+a d x^2\right )}{3 a^3 d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {c} (8 b+a c) \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 a^3 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (4 b+a c) \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{3 a^2 (b+a c) d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
-x*(d*x^2+c)/a/d/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+4/3*x*(a*d*x^2+a*c+b)/a ^2/d/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/3*(a*c+8*b)*x*(a*d*x^2+a*c+b)/a^3 /d/(d*x^2+c)/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/3*c^(3/2)*(a*c+4*b)*(a*d* x^2+a*c+b)*(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),(b/(a*c+b))^(1/2))/a^2/(a*c+b)/d^(3/2)/(d*x^2+c)/(( a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2 )+1/3*(a*c+8*b)*(a*d*x^2+a*c+b)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*El lipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(b/(a*c+b))^(1/2))*c^(1/2)/a^3 /d^(3/2)/(d*x^2+c)/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(c*(a*d*x^2+a*c+b)/(a *c+b)/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 10.41 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.62 \[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (c+d x^2\right ) \left (4 b+a \left (c+d x^2\right )\right )+i \left (8 b^2+9 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b (8 b+5 a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{3 a^3 d \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]
(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(c + d*x^2)*(4*b + a *(c + d*x^2)) + I*(8*b^2 + 9*a*b*c + a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a *c)] - I*b*(8*b + 5*a*c)*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x ^2)/c]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(3*a^3*d*Sqrt[ d/c]*(b + a*(c + d*x^2)))
Time = 0.62 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2058, 369, 403, 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.
| \(\displaystyle \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {x^2}{\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \int \frac {x^2 \left (d x^2+c\right )^{3/2}}{\left (a d x^2+b+a c\right )^{3/2}}dx}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\int \frac {\sqrt {d x^2+c} \left (4 d x^2+c\right )}{\sqrt {a d x^2+b+a c}}dx}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {\int -\frac {d \left ((8 b+a c) d x^2+c (4 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}+\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {\int \frac {d \left ((8 b+a c) d x^2+c (4 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {\int \frac {(8 b+a c) d x^2+c (4 b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {c (a c+4 b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+d (a c+8 b) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {d (a c+8 b) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {d (a c+8 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {a c+a d x^2+b} \left (\frac {\frac {4 x \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 a}-\frac {\frac {c^{3/2} (a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} (a c+b) \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+d (a c+8 b) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{3 a}}{a d}-\frac {x \left (c+d x^2\right )^{3/2}}{a d \sqrt {a c+a d x^2+b}}\right )}{\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}\) |
(Sqrt[b + a*c + a*d*x^2]*(-((x*(c + d*x^2)^(3/2))/(a*d*Sqrt[b + a*c + a*d* x^2])) + ((4*x*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/(3*a) - ((8*b + a* c)*d*((x*Sqrt[b + a*c + a*d*x^2])/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d ^(3/2)*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2) )])) + (c^(3/2)*(4*b + a*c)*Sqrt[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt [d]*x)/Sqrt[c]], b/(b + a*c)])/((b + a*c)*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c* (b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))/(3*a))/(a*d)))/(Sqrt[c + d *x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])
3.4.62.3.1 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[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_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, 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 = 9.20 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(\frac {\left (\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a \,d^{2} x^{5}+2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a c d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, b d \,x^{3}-\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,c^{2}+3 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, b d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} x +4 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c -8 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, b c x +3 \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, b c x \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{3 a^{2} d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a d \,x^{2}+a c +b \right )}\) | \(667\) |
| risch | \(\frac {x \left (a d \,x^{2}+a c +b \right )}{3 a^{2} d \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}-\frac {\left (-\frac {2 d \left (a c +5 b \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}+\frac {\left (a^{2} c^{2}+a b c -3 b^{2}\right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{a \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {3 b^{2} \left (a c +b \right ) \left (-\frac {\left (a \,d^{2} x^{2}+a c d \right ) x}{\left (a c +b \right ) b d \sqrt {\left (x^{2}+\frac {a c +b}{a d}\right ) \left (a \,d^{2} x^{2}+a c d \right )}}+\frac {\left (\frac {1}{a c +b}+\frac {a c}{\left (a c +b \right ) b}\right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 a d \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{b \left (a c +b \right ) \sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right )}{a}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{3 d \,a^{2} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(831\) |
1/3*(((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*d^2*x^5+2*(( a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*c*d*x^3+((a*d*x^2+a *c+b)*(d*x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*b*d*x^3-((a*d*x^2+a*c+b)*(d*x^ 2+c))^(1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE( x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*a*c^2+3*(-a*d/(a*c+b))^(1/2)*( a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*b*d*x^3+((a*d*x^2+a*c+b)*(d *x^2+c))^(1/2)*(-a*d/(a*c+b))^(1/2)*a*c^2*x+4*((a*d*x^2+a*c+b)*(d*x^2+c))^ (1/2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-a* d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*b*c-8*((a*d*x^2+a*c+b)*(d*x^2+c))^(1 /2)*((a*d*x^2+a*c+b)/(a*c+b))^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-a*d/ (a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*b*c+((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)* (-a*d/(a*c+b))^(1/2)*b*c*x+3*(-a*d/(a*c+b))^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b *d*x^2+a*c^2+b*c)^(1/2)*b*c*x)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^2/d/(a* d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(-a*d/(a*c+b))^(1/2)/(a*d*x^2 +a*c+b)
Time = 0.12 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {{\left ({\left (a^{2} c^{2} + 8 \, a b c\right )} d x^{3} + {\left (a^{2} c^{3} + 9 \, a b c^{2} + 8 \, b^{2} c\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left ({\left (a^{2} c + 4 \, a b\right )} d^{2} + {\left (a^{2} c^{2} + 8 \, a b c\right )} d\right )} x^{3} + {\left (a^{2} c^{3} + 9 \, a b c^{2} + 8 \, b^{2} c + {\left (a^{2} c^{2} + 5 \, a b c + 4 \, b^{2}\right )} d\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) + {\left (a^{2} d^{3} x^{6} + {\left (a^{2} c - 4 \, a b\right )} d^{2} x^{4} - a^{2} c^{3} - 9 \, a b c^{2} - {\left (a^{2} c^{2} + 13 \, a b c + 8 \, b^{2}\right )} d x^{2} - 8 \, b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{3 \, {\left (a^{4} d^{3} x^{3} + {\left (a^{4} c + a^{3} b\right )} d^{2} x\right )}} \]
1/3*(((a^2*c^2 + 8*a*b*c)*d*x^3 + (a^2*c^3 + 9*a*b*c^2 + 8*b^2*c)*x)*sqrt( a)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (((a^2*c + 4*a*b)*d^2 + (a^2*c^2 + 8*a*b*c)*d)*x^3 + (a^2*c^3 + 9*a*b*c^2 + 8*b^2* c + (a^2*c^2 + 5*a*b*c + 4*b^2)*d)*x)*sqrt(a)*sqrt(-c/d)*elliptic_f(arcsin (sqrt(-c/d)/x), (a*c + b)/(a*c)) + (a^2*d^3*x^6 + (a^2*c - 4*a*b)*d^2*x^4 - a^2*c^3 - 9*a*b*c^2 - (a^2*c^2 + 13*a*b*c + 8*b^2)*d*x^2 - 8*b^2*c)*sqrt ((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*d^3*x^3 + (a^4*c + a^3*b)*d^2*x)
\[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]