Integrand size = 22, antiderivative size = 171 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=-\frac {d (b c-15 a d) \sqrt {a+b x}}{4 a c^3 \sqrt {c+d x}}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}-\frac {(b c-5 a d) \sqrt {a+b x}}{4 a c^2 x \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}} \]
1/4*(-15*a^2*d^2+6*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^(3/2)/c^(7/2)-1/4*d*(-15*a*d+b*c)*(b*x+a)^(1/2)/a/c^3/(d* x+c)^(1/2)-1/2*(b*x+a)^(1/2)/c/x^2/(d*x+c)^(1/2)-1/4*(-5*a*d+b*c)*(b*x+a)^ (1/2)/a/c^2/x/(d*x+c)^(1/2)
Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-b c x (c+d x)+a \left (-2 c^2+5 c d x+15 d^2 x^2\right )\right )}{4 a c^3 x^2 \sqrt {c+d x}}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{3/2} c^{7/2}} \]
(Sqrt[a + b*x]*(-(b*c*x*(c + d*x)) + a*(-2*c^2 + 5*c*d*x + 15*d^2*x^2)))/( 4*a*c^3*x^2*Sqrt[c + d*x]) + ((b^2*c^2 + 6*a*b*c*d - 15*a^2*d^2)*ArcTanh[( Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*a^(3/2)*c^(7/2))
Time = 0.31 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {110, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\int \frac {b c-5 a d-4 b d x}{2 x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c-5 a d-4 b d x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {\int \frac {b^2 c^2+6 a b d c-15 a^2 d^2+2 b d (b c-5 a d) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {b^2 c^2+6 a b d c-15 a^2 d^2+2 b d (b c-5 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {-\frac {\frac {2 d \sqrt {a+b x} (b c-15 a d)}{c \sqrt {c+d x}}-\frac {2 \int -\frac {(b c-a d) \left (b^2 c^2+6 a b d c-15 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} (b c-15 a d)}{c \sqrt {c+d x}}}{2 a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {-\frac {\frac {2 \left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} (b c-15 a d)}{c \sqrt {c+d x}}}{2 a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {2 d \sqrt {a+b x} (b c-15 a d)}{c \sqrt {c+d x}}-\frac {2 \left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} (b c-5 a d)}{a c x \sqrt {c+d x}}}{4 c}-\frac {\sqrt {a+b x}}{2 c x^2 \sqrt {c+d x}}\) |
-1/2*Sqrt[a + b*x]/(c*x^2*Sqrt[c + d*x]) + (-(((b*c - 5*a*d)*Sqrt[a + b*x] )/(a*c*x*Sqrt[c + d*x])) - ((2*d*(b*c - 15*a*d)*Sqrt[a + b*x])/(c*Sqrt[c + d*x]) - (2*(b^2*c^2 + 6*a*b*c*d - 15*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*c)
3.6.90.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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_) )^(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. \(466\) vs. \(2(139)=278\).
Time = 0.53 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.73
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{3} x^{3}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{2} x^{3}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d \,x^{2}-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} x^{2}-30 a \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-10 a c d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b \,c^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a \,c^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 a \,c^{3} \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}}\) | \(467\) |
-1/8*(b*x+a)^(1/2)/a/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^2*d^3*x^3-6*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*c*d^2*x^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^2*c^2*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^2*c*d^2*x^2-6*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*c^2*d*x^2-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^2*c^3*x^2-30*a*d^2*x^2*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*b*c*d*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)-10*a*c*d*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*b*c^2*x*(a*c)^(1/2)* ((b*x+a)*(d*x+c))^(1/2)+4*a*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*c) ^(1/2)/x^2/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)
Time = 0.56 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.77 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\left [-\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}, -\frac {{\left ({\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d - 15 \, a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - 5 \, a^{2} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{2} c^{4} d x^{3} + a^{2} c^{5} x^{2}\right )}}\right ] \]
[-1/16*(((b^2*c^2*d + 6*a*b*c*d^2 - 15*a^2*d^3)*x^3 + (b^2*c^3 + 6*a*b*c^2 *d - 15*a^2*c*d^2)*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*c^3 + (a*b*c^2*d - 15*a^2* c*d^2)*x^2 + (a*b*c^3 - 5*a^2*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2* c^4*d*x^3 + a^2*c^5*x^2), -1/8*(((b^2*c^2*d + 6*a*b*c*d^2 - 15*a^2*d^3)*x^ 3 + (b^2*c^3 + 6*a*b*c^2*d - 15*a^2*c*d^2)*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*c^3 + (a*b*c^2*d - 15*a^2*c*d ^2)*x^2 + (a*b*c^3 - 5*a^2*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^4 *d*x^3 + a^2*c^5*x^2)]
\[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a + b x}}{x^{3} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1092 vs. \(2 (139) = 278\).
Time = 1.61 (sec) , antiderivative size = 1092, normalized size of antiderivative = 6.39 \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{3} {\left | b \right |}} + \frac {{\left (\sqrt {b d} b^{4} c^{2} + 6 \, \sqrt {b d} a b^{3} c d - 15 \, \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 b c^{3} {\left | b \right |}} - \frac {\sqrt {b d} b^{10} c^{5} - 11 \, \sqrt {b d} a b^{9} c^{4} d + 34 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{2} - 46 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{3} + 29 \, \sqrt {b d} a^{4} b^{6} c d^{4} - 7 \, \sqrt {b d} a^{5} b^{5} d^{5} - 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 )}^{2} b^{8} c^{4} + 28 \, \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 - 26 \, \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} - 20 \, \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} + 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 )}^{4} b^{6} c^{3} - 19 \, \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 - 11 \, \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} - \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} + 2 \, \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 c^{3} {\left | b \right |}} \]
2*sqrt(b*x + a)*b^2*d^2/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c^3*abs(b)) + 1/4*(sqrt(b*d)*b^4*c^2 + 6*sqrt(b*d)*a*b^3*c*d - 15*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*b*c^3*abs(b) ) - 1/2*(sqrt(b*d)*b^10*c^5 - 11*sqrt(b*d)*a*b^9*c^4*d + 34*sqrt(b*d)*a^2* b^8*c^3*d^2 - 46*sqrt(b*d)*a^3*b^7*c^2*d^3 + 29*sqrt(b*d)*a^4*b^6*c*d^4 - 7*sqrt(b*d)*a^5*b^5*d^5 - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2* c + (b*x + a)*b*d - a*b*d))^2*b^8*c^4 + 28*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 - 26*sqrt(b*d)*(s qrt(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 - 20*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(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 + 3*sqrt(b*d)*(sqrt(b*d)*sqr t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3 - 19*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 - 11*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 - sqrt(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 + 2*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^3*c*...
Timed out. \[ \int \frac {\sqrt {a+b x}}{x^3 (c+d x)^{3/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^3\,{\left (c+d\,x\right )}^{3/2}} \,d x \]