Integrand size = 21, antiderivative size = 249 \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {b c^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{d^3}-\frac {\left (5 b^2+60 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{48 a d^3}-\frac {(b+12 a c) \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 d^3}+\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{5/2}}{6 a d^3}-\frac {b \left (b^2+12 a b c-24 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{3/2} d^3} \]
1/6*(d*x^2+c)^3*((a*d*x^2+a*c+b)/(d*x^2+c))^(5/2)/a/d^3-1/16*b*(-24*a^2*c^ 2+12*a*b*c+b^2)*arctanh(((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a^(1/2))/a^(3/2) /d^3-b*c^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d^3-1/48*(-24*a^2*c^2+60*a*b* c+5*b^2)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/a/d^3-1/24*(12*a*c+b) *(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/d^3
Time = 0.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60 \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {a} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (3 b^2 \left (c+d x^2\right )-2 a b \left (47 c^2+16 c d x^2-7 d^2 x^4\right )+8 a^2 \left (c^3+d^3 x^6\right )\right )-3 b \left (b^2+12 a b c-24 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{48 a^{3/2} d^3} \]
(Sqrt[a]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(3*b^2*(c + d*x^2) - 2*a*b* (47*c^2 + 16*c*d*x^2 - 7*d^2*x^4) + 8*a^2*(c^3 + d^3*x^6)) - 3*b*(b^2 + 12 *a*b*c - 24*a^2*c^2)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a] ])/(48*a^(3/2)*d^3)
Time = 0.49 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2053, 2052, 27, 366, 360, 1471, 27, 299, 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.
| \(\displaystyle \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int x^5 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle -b d \int \frac {x^8 \left (-c x^4+b+a c\right )^2}{d^4 \left (a-x^4\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \int \frac {x^8 \left (-c x^4+b+a c\right )^2}{\left (a-x^4\right )^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{d^3}\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\int \frac {x^8 \left (6 a c^2 x^4+5 b^2-6 (b+a c)^2\right )}{\left (a-x^4\right )^3}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{6 a}\right )}{d^3}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\frac {1}{4} \int \frac {-24 a c^2 x^8+4 b (b+12 a c) x^4+a b (b+12 a c)}{\left (a-x^4\right )^2}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}-\frac {a b (12 a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 \left (a-x^4\right )^2}}{6 a}\right )}{d^3}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\frac {1}{4} \left (\frac {\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a-x^4\right )}-\frac {\int \frac {3 a \left (-16 a c^2 x^4+b^2-8 a^2 c^2+12 a b c\right )}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 a}\right )-\frac {a b (12 a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 \left (a-x^4\right )^2}}{6 a}\right )}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\frac {1}{4} \left (\frac {\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a-x^4\right )}-\frac {3}{2} \int \frac {-16 a c^2 x^4+b^2-8 a^2 c^2+12 a b c}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\right )-\frac {a b (12 a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 \left (a-x^4\right )^2}}{6 a}\right )}{d^3}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\frac {1}{4} \left (\frac {\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a-x^4\right )}-\frac {3}{2} \left (\left (-24 a^2 c^2+12 a b c+b^2\right ) \int \frac {1}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}+16 a c^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}\right )\right )-\frac {a b (12 a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 \left (a-x^4\right )^2}}{6 a}\right )}{d^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {b \left (\frac {b^2 x^{10}}{6 a \left (a-x^4\right )^3}-\frac {\frac {1}{4} \left (\frac {\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a-x^4\right )}-\frac {3}{2} \left (\frac {\left (-24 a^2 c^2+12 a b c+b^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{\sqrt {a}}+16 a c^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}\right )\right )-\frac {a b (12 a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 \left (a-x^4\right )^2}}{6 a}\right )}{d^3}\) |
-((b*((b^2*x^10)/(6*a*(a - x^4)^3) - (-1/4*(a*b*(b + 12*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(a - x^4)^2 + (((5*b^2 + 60*a*b*c - 24*a^2*c^2)* Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(2*(a - x^4)) - (3*(16*a*c^2*Sqrt[( b + a*c + a*d*x^2)/(c + d*x^2)] + ((b^2 + 12*a*b*c - 24*a^2*c^2)*ArcTanh[S qrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/Sqrt[a]))/2)/4)/(6*a)))/d^3 )
3.4.31.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
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 = 0.22 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\left (8 a^{2} d^{2} x^{4}-8 a^{2} c d \,x^{2}+14 a b d \,x^{2}+8 a^{2} c^{2}-46 a b c +3 b^{2}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 a \,d^{3}}+\frac {b \left (-\frac {16 a \,c^{2} \left (a d \,x^{2}+a c +b \right )}{d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {\left (24 a^{2} c^{2}-12 a b c -b^{2}\right ) \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{2 \sqrt {a \,d^{2}}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{16 a \,d^{2} \left (a d \,x^{2}+a c +b \right )}\) | \(295\) |
| default | \(\text {Expression too large to display}\) | \(1018\) |
1/48/a/d^3*(8*a^2*d^2*x^4-8*a^2*c*d*x^2+14*a*b*d*x^2+8*a^2*c^2-46*a*b*c+3* b^2)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/16*b/a/d^2*(-16*a*c^2*( a*d*x^2+a*c+b)/d/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+1/2*(24*a ^2*c^2-12*a*b*c-b^2)*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))*((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)/(a*d*x^2+a*c+b)
Time = 0.41 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.71 \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\left [\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{2} d^{3}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{2} d^{3}}\right ] \]
[1/192*(3*(24*a^2*b*c^2 - 12*a*b^2*c - b^3)*sqrt(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*(8*a^3*d^3*x^6 + 14*a^2*b*d^2*x^4 + 8*a^3*c^3 - 94*a^2*b*c^2 + 3*a* b^2*c - (32*a^2*b*c - 3*a*b^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c) ))/(a^2*d^3), -1/96*(3*(24*a^2*b*c^2 - 12*a*b^2*c - b^3)*sqrt(-a)*arctan(1 /2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/ (a^2*d*x^2 + a^2*c + a*b)) - 2*(8*a^3*d^3*x^6 + 14*a^2*b*d^2*x^4 + 8*a^3*c ^3 - 94*a^2*b*c^2 + 3*a*b^2*c - (32*a^2*b*c - 3*a*b^2)*d*x^2)*sqrt((a*d*x^ 2 + a*c + b)/(d*x^2 + c)))/(a^2*d^3)]
\[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^{5} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.48 \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {b c^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d^{3}} - \frac {3 \, {\left (8 \, a^{2} b c^{2} - 20 \, a b^{2} c + b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{2} - 12 \, a^{2} b^{2} c - a b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{2} - 12 \, a^{3} b^{2} c - a^{2} b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} d^{3} - \frac {3 \, {\left (a d x^{2} + a c + b\right )} a^{3} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} a^{2} d^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (a d x^{2} + a c + b\right )}^{3} a d^{3}}{{\left (d x^{2} + c\right )}^{3}}\right )}} - \frac {{\left (24 \, a^{2} c^{2} - 12 \, a b c - b^{2}\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{32 \, a^{\frac {3}{2}} d^{3}} \]
-b*c^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/d^3 - 1/48*(3*(8*a^2*b*c^2 - 20*a*b^2*c + b^3)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(5/2) - 8*(6*a^3*b*c^2 - 12*a^2*b^2*c - a*b^3)*((a*d*x^2 + a*c + b)/(d*x^2 + c))^(3/2) + 3*(8*a^ 4*b*c^2 - 12*a^3*b^2*c - a^2*b^3)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/( a^4*d^3 - 3*(a*d*x^2 + a*c + b)*a^3*d^3/(d*x^2 + c) + 3*(a*d*x^2 + a*c + b )^2*a^2*d^3/(d*x^2 + c)^2 - (a*d*x^2 + a*c + b)^3*a*d^3/(d*x^2 + c)^3) - 1 /32*(24*a^2*c^2 - 12*a*b*c - b^2)*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^(3/ 2)*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (229) = 458\).
Time = 0.80 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.08 \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {1}{48} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, {\left (\frac {4 \, a x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d} - \frac {4 \, a^{3} c d^{6} \mathrm {sgn}\left (d x^{2} + c\right ) - 7 \, a^{2} b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} x^{2} + \frac {8 \, a^{3} c^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) - 46 \, a^{2} b c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, a b^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} - \frac {{\left (24 \, a^{2} b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 12 \, a b^{2} c \mathrm {sgn}\left (d x^{2} + c\right ) - b^{3} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | 2 \, a^{2} c^{3} d + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{\frac {3}{2}} c^{2} {\left | d \right |} + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a c d + a b c^{2} d + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} \sqrt {a} {\left | d \right |} + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} b c {\left | d \right |} + {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} b d \right |}\right )}{96 \, a^{\frac {3}{2}} d^{2} {\left | d \right |}} \]
1/48*sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c)*(2*(4*a*x^2*sgn (d*x^2 + c)/d - (4*a^3*c*d^6*sgn(d*x^2 + c) - 7*a^2*b*d^6*sgn(d*x^2 + c))/ (a^2*d^8))*x^2 + (8*a^3*c^2*d^5*sgn(d*x^2 + c) - 46*a^2*b*c*d^5*sgn(d*x^2 + c) + 3*a*b^2*d^5*sgn(d*x^2 + c))/(a^2*d^8)) - 1/96*(24*a^2*b*c^2*sgn(d*x ^2 + c) - 12*a*b^2*c*sgn(d*x^2 + c) - b^3*sgn(d*x^2 + c))*log(abs(2*a^2*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^(3/2)*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*c*d + a*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*sqrt(a)*abs(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) )*sqrt(a)*b*c*abs(d) + (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*b*d))/(a^(3/2)*d^2*abs(d))
Timed out. \[ \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\int x^5\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]