Integrand size = 22, antiderivative size = 194 \[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {d (3 b c-5 a d) (b c+3 a d) \sqrt {a+b x}}{4 a^2 c^3 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}+\frac {(3 b c+5 a d) \sqrt {a+b x}}{4 a^2 c^2 x \sqrt {c+d x}}-\frac {3 \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2}} \]
-3/4*(5*a^2*d^2+2*a*b*c*d+b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/( d*x+c)^(1/2))/a^(5/2)/c^(7/2)+1/4*d*(-5*a*d+3*b*c)*(3*a*d+b*c)*(b*x+a)^(1/ 2)/a^2/c^3/(-a*d+b*c)/(d*x+c)^(1/2)-1/2*(b*x+a)^(1/2)/a/c/x^2/(d*x+c)^(1/2 )+1/4*(5*a*d+3*b*c)*(b*x+a)^(1/2)/a^2/c^2/x/(d*x+c)^(1/2)
Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \left (3 b^2 c^2 x (c+d x)+a^2 d \left (2 c^2-5 c d x-15 d^2 x^2\right )+2 a b c \left (-c^2+c d x+2 d^2 x^2\right )\right )}{x^2 \sqrt {c+d x}}-3 \left (b^3 c^3+a b^2 c^2 d+3 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{7/2} (b c-a d)} \]
((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*(3*b^2*c^2*x*(c + d*x) + a^2*d*(2*c^2 - 5* c*d*x - 15*d^2*x^2) + 2*a*b*c*(-c^2 + c*d*x + 2*d^2*x^2)))/(x^2*Sqrt[c + d *x]) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt [c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(5/2)*c^(7/2)*(b*c - a*d ))
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {114, 27, 168, 27, 169, 27, 104, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {3 b c+5 a d+4 b d x}{2 x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {3 b c+5 a d+4 b d x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (b^2 c^2+2 a b d c+5 a^2 d^2\right )+2 b d (3 b c+5 a d) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (b^2 c^2+2 a b d c+5 a^2 d^2\right )+2 b d (3 b c+5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {2 d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {3 (b c-a d) \left (b^2 c^2+2 a b d c+5 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{c \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {\frac {6 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{c \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {2 d \sqrt {a+b x} (3 b c-5 a d) (3 a d+b c)}{c \sqrt {c+d x} (b c-a d)}-\frac {6 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} (5 a d+3 b c)}{a c x \sqrt {c+d x}}}{4 a c}-\frac {\sqrt {a+b x}}{2 a c x^2 \sqrt {c+d x}}\) |
-1/2*Sqrt[a + b*x]/(a*c*x^2*Sqrt[c + d*x]) - (-(((3*b*c + 5*a*d)*Sqrt[a + b*x])/(a*c*x*Sqrt[c + d*x])) - ((2*d*(3*b*c - 5*a*d)*(b*c + 3*a*d)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (6*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^ 2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3 /2)))/(2*a*c))/(4*a*c)
3.8.47.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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. \(682\) vs. \(2(162)=324\).
Time = 0.57 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.52
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{4} x^{3}-9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{3} x^{3}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d^{2} x^{3}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{3} x^{2}-9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d^{2} x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} d \,x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{4} x^{2}-30 a^{2} d^{3} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+8 a b c \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 b^{2} c^{2} d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-10 a^{2} c \,d^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a b \,c^{2} d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 b^{2} c^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a^{2} c^{2} d \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-4 a b \,c^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 a^{2} c^{3} \left (a d -b c \right ) \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}}\) | \(683\) |
-1/8*(b*x+a)^(1/2)/a^2/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x +c))^(1/2)+2*a*c)/x)*a^3*d^4*x^3-9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)* (d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c*d^3*x^3-3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*( (b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^2*c^2*d^2*x^3-3*ln((a*d*x+b*c*x+2*(a* c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*d*x^3+15*ln((a*d*x+b*c* x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*c*d^3*x^2-9*ln((a*d* x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c^2*d^2*x^2- 3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b^2*c^ 3*d*x^2-3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)* b^3*c^4*x^2-30*a^2*d^3*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+8*a*b*c*d^2 *x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*b^2*c^2*d*x^2*(a*c)^(1/2)*((b*x +a)*(d*x+c))^(1/2)-10*a^2*c*d^2*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a* b*c^2*d*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*b^2*c^3*x*(a*c)^(1/2)*((b* x+a)*(d*x+c))^(1/2)+4*a^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-4*a*b* c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*d-b*c)/(a*c)^(1/2)/x^2/((b*x+a )*(d*x+c))^(1/2)/(d*x+c)^(1/2)
Time = 0.71 (sec) , antiderivative size = 664, normalized size of antiderivative = 3.42 \[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} + {\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} b c^{4} - 2 \, a^{3} c^{3} d - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{2} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left ({\left (a^{3} b c^{5} d - a^{4} c^{4} d^{2}\right )} x^{3} + {\left (a^{3} b c^{6} - a^{4} c^{5} d\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} + {\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} b c^{4} - 2 \, a^{3} c^{3} d - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{2} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left ({\left (a^{3} b c^{5} d - a^{4} c^{4} d^{2}\right )} x^{3} + {\left (a^{3} b c^{6} - a^{4} c^{5} d\right )} x^{2}\right )}}\right ] \]
[1/16*(3*((b^3*c^3*d + a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - 5*a^3*d^4)*x^3 + (b ^3*c^4 + a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x^2)*sqrt(a*c)*log(( 8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x )*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(2*a^2*b*c^4 - 2*a^3*c^3*d - (3*a*b^2*c^3*d + 4*a^2*b*c^2*d^2 - 15*a^3*c *d^3)*x^2 - (3*a*b^2*c^4 + 2*a^2*b*c^3*d - 5*a^3*c^2*d^2)*x)*sqrt(b*x + a) *sqrt(d*x + c))/((a^3*b*c^5*d - a^4*c^4*d^2)*x^3 + (a^3*b*c^6 - a^4*c^5*d) *x^2), 1/8*(3*((b^3*c^3*d + a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - 5*a^3*d^4)*x^3 + (b^3*c^4 + a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x^2)*sqrt(-a*c) *arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) /(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(2*a^2*b*c^4 - 2*a^3 *c^3*d - (3*a*b^2*c^3*d + 4*a^2*b*c^2*d^2 - 15*a^3*c*d^3)*x^2 - (3*a*b^2*c ^4 + 2*a^2*b*c^3*d - 5*a^3*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3* b*c^5*d - a^4*c^4*d^2)*x^3 + (a^3*b*c^6 - a^4*c^5*d)*x^2)]
\[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (162) = 324\).
Time = 1.10 (sec) , antiderivative size = 1105, normalized size of antiderivative = 5.70 \[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b c^{4} {\left | b \right |} - a c^{3} d {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {3 \, {\left (\sqrt {b d} b^{4} c^{2} + 2 \, \sqrt {b d} a b^{3} c d + 5 \, \sqrt {b d} a^{2} b^{2} d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{4 \, \sqrt {-a b c d} a^{2} b c^{3} {\left | b \right |}} + \frac {3 \, \sqrt {b d} b^{10} c^{5} - 5 \, \sqrt {b d} a b^{9} c^{4} d - 10 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} + 30 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} - 25 \, \sqrt {b d} a^{4} b^{6} c d^{4} + 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{8} c^{4} - 16 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{3} d + 38 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{2} d^{2} + 8 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c d^{3} - 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} d^{4} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{6} c^{3} + 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{2} d + 23 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c d^{2} + 21 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} d^{3} - 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{4} c^{2} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{3} c d - 7 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} d^{2}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a^{2} c^{3} {\left | b \right |}} \]
-2*sqrt(b*x + a)*b^2*d^3/((b*c^4*abs(b) - a*c^3*d*abs(b))*sqrt(b^2*c + (b* x + a)*b*d - a*b*d)) - 3/4*(sqrt(b*d)*b^4*c^2 + 2*sqrt(b*d)*a*b^3*c*d + 5* sqrt(b*d)*a^2*b^2*d^2)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a* b*c*d)*a^2*b*c^3*abs(b)) + 1/2*(3*sqrt(b*d)*b^10*c^5 - 5*sqrt(b*d)*a*b^9*c ^4*d - 10*sqrt(b*d)*a^2*b^8*c^3*d^2 + 30*sqrt(b*d)*a^3*b^7*c^2*d^3 - 25*sq rt(b*d)*a^4*b^6*c*d^4 + 7*sqrt(b*d)*a^5*b^5*d^5 - 9*sqrt(b*d)*(sqrt(b*d)*s qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^4 - 16*sqrt(b* d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^7 *c^3*d + 38*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^2*a^2*b^6*c^2*d^2 + 8*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*c*d^3 - 21*sqrt(b*d)*(sqrt(b*d )*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^4*d^4 + 9*s qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4 *b^6*c^3 + 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^4*a*b^5*c^2*d + 23*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt (b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^4*c*d^2 + 21*sqrt(b*d)*(sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*d^3 - 3*sq rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6* b^4*c^2 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)...
Timed out. \[ \int \frac {1}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^3\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]