Integrand size = 22, antiderivative size = 174 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \]
-(a*d+5*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/ d^(7/2)-2/3*c*x^2*(b*x+a)^(1/2)/d/(-a*d+b*c)/(d*x+c)^(3/2)+1/3*(c*(3*a^2*d ^2-22*a*b*c*d+15*b^2*c^2)+d*(-3*a*d+5*b*c)*(-a*d+b*c)*x)*(b*x+a)^(1/2)/b/d ^3/(-a*d+b*c)^2/(d*x+c)^(1/2)
Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 d^2 (c+d x)^2-2 a b c d \left (11 c^2+15 c d x+3 d^2 x^2\right )+b^2 c^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 b d^3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \]
(Sqrt[a + b*x]*(3*a^2*d^2*(c + d*x)^2 - 2*a*b*c*d*(11*c^2 + 15*c*d*x + 3*d ^2*x^2) + b^2*c^2*(15*c^2 + 20*c*d*x + 3*d^2*x^2)))/(3*b*d^3*(b*c - a*d)^2 *(c + d*x)^(3/2)) - ((5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b ]*Sqrt[c + d*x])])/(b^(3/2)*d^(7/2))
Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {109, 27, 160, 66, 221}
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.
| \(\displaystyle \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \int \frac {x (4 a c+(5 b c-3 a d) x)}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x (4 a c+(5 b c-3 a d) x)}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 160 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {3 (b c-a d) (a d+5 b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b d^2}}{3 d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {3 (b c-a d) (a d+5 b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b d^2}}{3 d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {3 (b c-a d) (a d+5 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}}{3 d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
(-2*c*x^2*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) + ((Sqrt[a + b* x]*(c*(15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2) + d*(5*b*c - 3*a*d)*(b*c - a*d )*x))/(b*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (3*(b*c - a*d)*(5*b*c + a*d)*Arc Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*d^(5/2)))/ (3*d*(b*c - a*d))
3.8.49.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* (f*g + e*h) - c*f*h*(m + 2)))/(b^2*d) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(927\) vs. \(2(150)=300\).
Time = 1.67 (sec) , antiderivative size = 928, normalized size of antiderivative = 5.33
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{5} x^{2}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{4} x^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d^{3} x^{2}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3} d^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} c \,d^{4} x +18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,c^{2} d^{3} x -54 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{3} d^{2} x +30 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{4} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{4} x^{2}+12 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} c^{2} d^{3}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,c^{3} d^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{4} d +15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{5}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{3} x +60 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -40 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c^{2} d^{2}+44 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right )}{6 \sqrt {b d}\, b \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\) | \(928\) |
-1/6*(b*x+a)^(1/2)*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2 )+a*d+b*c)/(b*d)^(1/2))*a^3*d^5*x^2+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^ (1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b*c*d^4*x^2-27*ln(1/2*(2*b*d*x +2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^2*c^2*d^3 *x^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b* d)^(1/2))*b^3*c^3*d^2*x^2+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d )^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*c*d^4*x+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d *x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b*c^2*d^3*x-54*ln(1/2*( 2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^2* c^3*d^2*x+30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c )/(b*d)^(1/2))*b^3*c^4*d*x-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*d^4*x ^2+12*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d^3*x^2-6*(b*d)^(1/2)*((b* x+a)*(d*x+c))^(1/2)*b^2*c^2*d^2*x^2+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^ (1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*c^2*d^3+9*ln(1/2*(2*b*d*x+2*(( b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b*c^3*d^2-27*l n(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2)) *a*b^2*c^4*d+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+ b*c)/(b*d)^(1/2))*b^3*c^5-12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*c*d^3 *x+60*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x-40*(b*d)^(1/2)*((b *x+a)*(d*x+c))^(1/2)*b^2*c^3*d*x-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (150) = 300\).
Time = 0.38 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.15 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}\right ] \]
[1/12*(3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (5*b ^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(5*b^3*c^4 *d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x)*sqrt(b*d)*log(8*b^2 *d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b* d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(15*b^3*c^4* d - 22*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + 3*(b^3*c^2*d^3 - 2*a*b^2*c*d^4 + a^2*b*d^5)*x^2 + 2*(10*b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4)*x)* sqrt(b*x + a)*sqrt(d*x + c))/(b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 + a^2*b^2*c^2* d^6 + (b^4*c^2*d^6 - 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*(b^4*c^3*d^5 - 2 *a*b^3*c^2*d^6 + a^2*b^2*c*d^7)*x), 1/6*(3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3* a^2*b*c^3*d^2 + a^3*c^2*d^3 + (5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c *d^4 + a^3*d^5)*x^2 + 2*(5*b^3*c^4*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*d^4)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt (b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(15*b^3*c^4*d - 22*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + 3*(b^3*c^2*d^3 - 2 *a*b^2*c*d^4 + a^2*b*d^5)*x^2 + 2*(10*b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 3*a ^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 + a^2*b^2*c^2*d^6 + (b^4*c^2*d^6 - 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*( b^4*c^3*d^5 - 2*a*b^3*c^2*d^6 + a^2*b^2*c*d^7)*x)]
\[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{3}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (150) = 300\).
Time = 0.37 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.14 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{5} c d^{5} {\left | b \right |} + a^{2} b^{4} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac {2 \, {\left (10 \, b^{7} c^{3} d^{3} {\left | b \right |} - 18 \, a b^{6} c^{2} d^{4} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{5} {\left | b \right |} - 3 \, a^{3} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{4} d^{2} {\left | b \right |} - 14 \, a b^{7} c^{3} d^{3} {\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{5} {\left | b \right |} + a^{4} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b c {\left | b \right |} + a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}} \]
1/3*((b*x + a)*(3*(b^6*c^2*d^4*abs(b) - 2*a*b^5*c*d^5*abs(b) + a^2*b^4*d^6 *abs(b))*(b*x + a)/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7) + 2*(10*b^7 *c^3*d^3*abs(b) - 18*a*b^6*c^2*d^4*abs(b) + 9*a^2*b^5*c*d^5*abs(b) - 3*a^3 *b^4*d^6*abs(b))/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)) + 3*(5*b^8*c ^4*d^2*abs(b) - 14*a*b^7*c^3*d^3*abs(b) + 12*a^2*b^6*c^2*d^4*abs(b) - 4*a^ 3*b^5*c*d^5*abs(b) + a^4*b^4*d^6*abs(b))/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^ 2*b^5*d^7))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + (5*b*c*a bs(b) + a*d*abs(b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3)
Timed out. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^3}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]