Integrand size = 21, antiderivative size = 187 \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b (b+a c)}{a^3 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(7 b+4 a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{8 a^3 d^2}+\frac {\left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{4 a^2 d^2}+\frac {3 b (5 b+4 a c) \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{8 a^{7/2} d^2} \]
3/8*b*(4*a*c+5*b)*arctanh(((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^(1/2))/a^(7/ 2)/d^2-b*(a*c+b)/a^3/d^2/((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)-1/8*(4*a*c+7*b) *(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^3/d^2+1/4*(d*x^2+c)^2*((a*d *x^2+a*c+b)/(d*x^2+c))^(1/2)/a^2/d^2
Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {a} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (15 b^2+a b \left (17 c+5 d x^2\right )+2 a^2 \left (c^2-d^2 x^4\right )\right )}{b+a \left (c+d x^2\right )}+3 b (5 b+4 a c) \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{8 a^{7/2} d^2} \]
(-((Sqrt[a]*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(15*b^2 + a* b*(17*c + 5*d*x^2) + 2*a^2*(c^2 - d^2*x^4)))/(b + a*(c + d*x^2))) + 3*b*(5 *b + 4*a*c)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/(8*a^( 7/2)*d^2)
Time = 0.42 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2057, 2053, 2052, 25, 27, 361, 25, 27, 361, 25, 359, 219}
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.
| \(\displaystyle \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {x^3}{\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int -\frac {-c x^4+b+a c}{d^3 x^4 \left (a-x^4\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b d \int \frac {-c x^4+b+a c}{d^3 x^4 \left (a-x^4\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {-c x^4+b+a c}{x^4 \left (a-x^4\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{d^2}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {b \left (\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}-\frac {1}{4} \int -\frac {3 b x^4+4 a (b+a c)}{a^2 x^4 \left (a-x^4\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {1}{4} \int \frac {3 b x^4+4 a (b+a c)}{a^2 x^4 \left (a-x^4\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {3 b x^4+4 a (b+a c)}{x^4 \left (a-x^4\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 a^2}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {b \left (\frac {\frac {(4 a c+7 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 a \left (a-x^4\right )}-\frac {1}{2} \int -\frac {\left (\frac {7 b}{a}+4 c\right ) x^4+8 (b+a c)}{x^4 \left (a-x^4\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 a^2}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {1}{2} \int \frac {\left (\frac {7 b}{a}+4 c\right ) x^4+8 (b+a c)}{x^4 \left (a-x^4\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}+\frac {(4 a c+7 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 a \left (a-x^4\right )}}{4 a^2}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {b \left (\frac {\frac {1}{2} \left (\frac {3 (4 a c+5 b) \int \frac {1}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{a}-\frac {8 (a c+b)}{a x^2}\right )+\frac {(4 a c+7 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 a \left (a-x^4\right )}}{4 a^2}+\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}\right )}{d^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {b \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 \left (a-x^4\right )^2}+\frac {\frac {1}{2} \left (\frac {3 (4 a c+5 b) \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {8 (a c+b)}{a x^2}\right )+\frac {(4 a c+7 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 a \left (a-x^4\right )}}{4 a^2}\right )}{d^2}\) |
(b*((b*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(4*a^2*(a - x^4)^2) + (((7*b + 4*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(2*a*(a - x^4)) + ((-8*(b + a*c))/(a*x^2) + (3*(5*b + 4*a*c)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/a^(3/2))/2)/(4*a^2)))/d^2
3.4.56.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Time = 0.20 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(-\frac {\left (-2 a d \,x^{2}+2 a c +7 b \right ) \left (a d \,x^{2}+a c +b \right )}{8 d^{2} a^{3} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {b \left (\frac {\left (12 a c +15 b \right ) \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{2 \sqrt {a \,d^{2}}}-\frac {8 \left (a c +b \right ) \left (d \,x^{2}+c \right )}{d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{8 a^{3} d \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(248\) |
| default | \(-\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} d^{2} x^{4}-12 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} b c \,d^{2} x^{2}-15 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a \,b^{2} d^{2} x^{2}+10 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b d \,x^{2}-12 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} b \,c^{2} d +4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c^{2}-27 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a \,b^{2} c d +16 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, a b c +18 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b c -15 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{3} d +16 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, b^{2}+14 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2}\right )}{16 a^{3} d^{2} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a \,d^{2}}\, \left (a d \,x^{2}+a c +b \right )}\) | \(783\) |
-1/8/d^2*(-2*a*d*x^2+2*a*c+7*b)*(a*d*x^2+a*c+b)/a^3/((a*d*x^2+a*c+b)/(d*x^ 2+c))^(1/2)+1/8*b/a^3/d*(1/2*(12*a*c+15*b)*ln((a*c*d+1/2*b*d+a*d^2*x^2)/(a *d^2)^(1/2)+(a*c^2+b*c+(2*a*c*d+b*d)*x^2+a*d^2*x^4)^(1/2))/(a*d^2)^(1/2)-8 *(a*c+b)*(d*x^2+c)/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*x^2+a*c+b)/(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(d*x^2+c)
Time = 0.36 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.89 \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (4 \, a^{2} b c^{2} + 9 \, a b^{2} c + {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} d x^{2} + 5 \, b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (2 \, a^{3} d^{3} x^{6} + {\left (2 \, a^{3} c - 5 \, a^{2} b\right )} d^{2} x^{4} - 2 \, a^{3} c^{3} - 17 \, a^{2} b c^{2} - 15 \, a b^{2} c - {\left (2 \, a^{3} c^{2} + 22 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{5} d^{3} x^{2} + {\left (a^{5} c + a^{4} b\right )} d^{2}\right )}}, -\frac {3 \, {\left (4 \, a^{2} b c^{2} + 9 \, a b^{2} c + {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} d x^{2} + 5 \, b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (2 \, a^{3} d^{3} x^{6} + {\left (2 \, a^{3} c - 5 \, a^{2} b\right )} d^{2} x^{4} - 2 \, a^{3} c^{3} - 17 \, a^{2} b c^{2} - 15 \, a b^{2} c - {\left (2 \, a^{3} c^{2} + 22 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{5} d^{3} x^{2} + {\left (a^{5} c + a^{4} b\right )} d^{2}\right )}}\right ] \]
[1/32*(3*(4*a^2*b*c^2 + 9*a*b^2*c + (4*a^2*b*c + 5*a*b^2)*d*x^2 + 5*b^3)*s qrt(a)*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a*b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*sqrt((a *d*x^2 + a*c + b)/(d*x^2 + c))) + 4*(2*a^3*d^3*x^6 + (2*a^3*c - 5*a^2*b)*d ^2*x^4 - 2*a^3*c^3 - 17*a^2*b*c^2 - 15*a*b^2*c - (2*a^3*c^2 + 22*a^2*b*c + 15*a*b^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^5*d^3*x^2 + (a ^5*c + a^4*b)*d^2), -1/16*(3*(4*a^2*b*c^2 + 9*a*b^2*c + (4*a^2*b*c + 5*a*b ^2)*d*x^2 + 5*b^3)*sqrt(-a)*arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)*sq rt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) - 2*(2*a^3* d^3*x^6 + (2*a^3*c - 5*a^2*b)*d^2*x^4 - 2*a^3*c^3 - 17*a^2*b*c^2 - 15*a*b^ 2*c - (2*a^3*c^2 + 22*a^2*b*c + 15*a*b^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/ (d*x^2 + c)))/(a^5*d^3*x^2 + (a^5*c + a^4*b)*d^2)]
\[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.40 \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {8 \, a^{3} b c + 8 \, a^{2} b^{2} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} {\left (4 \, a b c + 5 \, b^{2}\right )}}{{\left (d x^{2} + c\right )}^{2}} - \frac {5 \, {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}}{8 \, {\left (a^{5} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - 2 \, a^{4} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + a^{3} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}}\right )}} - \frac {3 \, {\left (4 \, a c + 5 \, b\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{16 \, a^{\frac {7}{2}} d^{2}} \]
-1/8*(8*a^3*b*c + 8*a^2*b^2 + 3*(a*d*x^2 + a*c + b)^2*(4*a*b*c + 5*b^2)/(d *x^2 + c)^2 - 5*(4*a^2*b*c + 5*a*b^2)*(a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^ 5*d^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - 2*a^4*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + a^3*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(5/2)) - 3/16*(4*a*c + 5*b)*b*log(-(sqrt(a) - sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c) ))/(sqrt(a) + sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))))/(a^(7/2)*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (169) = 338\).
Time = 0.61 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.95 \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {1}{8} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (\frac {2 \, x^{2}}{a^{2} d \mathrm {sgn}\left (d x^{2} + c\right )} - \frac {2 \, a^{6} c d^{2} + 7 \, a^{5} b d^{2}}{a^{8} d^{4} \mathrm {sgn}\left (d x^{2} + c\right )}\right )} - \frac {{\left (4 \, a b c + 5 \, b^{2}\right )} \log \left ({\left | 2 \, a^{3} c^{3} d + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{\frac {5}{2}} c^{2} {\left | d \right |} + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{2} c d + 5 \, a^{2} b c^{2} d + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a^{\frac {3}{2}} {\left | d \right |} + 10 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{\frac {3}{2}} b c {\left | d \right |} + 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a b d + 4 \, a b^{2} c d + 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} b^{2} {\left | d \right |} + b^{3} d \right |}\right )}{16 \, a^{\frac {7}{2}} d {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )} - \frac {{\left (4 \, a^{\frac {9}{2}} b c d^{2} {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right ) + 5 \, a^{\frac {7}{2}} b^{2} d^{2} {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | a \right |}\right )}{16 \, a^{7} d^{5}} \]
1/8*sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c)*(2*x^2/(a^2*d*sg n(d*x^2 + c)) - (2*a^6*c*d^2 + 7*a^5*b*d^2)/(a^8*d^4*sgn(d*x^2 + c))) - 1/ 16*(4*a*b*c + 5*b^2)*log(abs(2*a^3*c^3*d + 6*(sqrt(a*d^2)*x^2 - sqrt(a*d^2 *x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^(5/2)*c^2*abs(d) + 6*(sqrt( a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a^2* c*d + 5*a^2*b*c^2*d + 2*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*a^(3/2)*abs(d) + 10*(sqrt(a*d^2)*x^2 - sqrt(a*d^ 2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))*a^(3/2)*b*c*abs(d) + 5*(sqrt (a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a*b *d + 4*a*b^2*c*d + 4*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d *x^2 + a*c^2 + b*c))*sqrt(a)*b^2*abs(d) + b^3*d))/(a^(7/2)*d*abs(d)*sgn(d* x^2 + c)) - 1/16*(4*a^(9/2)*b*c*d^2*abs(d)*sgn(d*x^2 + c) + 5*a^(7/2)*b^2* d^2*abs(d)*sgn(d*x^2 + c))*log(abs(a))/(a^7*d^5)
Timed out. \[ \int \frac {x^3}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]