Integrand size = 22, antiderivative size = 148 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {5 d \sqrt {a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}-\frac {d (13 b c-15 a d) \sqrt {a+b x}}{3 c^3 (b c-a d) \sqrt {c+d x}}-\frac {(b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}} \]
-(-5*a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(7/2) /a^(1/2)-5/3*d*(b*x+a)^(1/2)/c^2/(d*x+c)^(3/2)-(b*x+a)^(1/2)/c/x/(d*x+c)^( 3/2)-1/3*d*(-15*a*d+13*b*c)*(b*x+a)^(1/2)/c^3/(-a*d+b*c)/(d*x+c)^(1/2)
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-b c \left (3 c^2+18 c d x+13 d^2 x^2\right )+a d \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )}{3 c^3 (b c-a d) x (c+d x)^{3/2}}+\frac {(-b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{7/2}} \]
(Sqrt[a + b*x]*(-(b*c*(3*c^2 + 18*c*d*x + 13*d^2*x^2)) + a*d*(3*c^2 + 20*c *d*x + 15*d^2*x^2)))/(3*c^3*(b*c - a*d)*x*(c + d*x)^(3/2)) + ((-(b*c) + 5* a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^ (7/2))
Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {110, 27, 169, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\int \frac {b c-5 a d-4 b d x}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c-5 a d-4 b d x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {(b c-a d) (3 (b c-5 a d)-10 b d x)}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 (b c-5 a d)-10 b d x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {3 (b c-5 a d) (b c-a d)}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}-\frac {2 d \sqrt {a+b x} (13 b c-15 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 (b c-5 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}-\frac {2 d \sqrt {a+b x} (13 b c-15 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\frac {\frac {6 (b c-5 a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {2 d \sqrt {a+b x} (13 b c-15 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {-\frac {6 (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {2 d \sqrt {a+b x} (13 b c-15 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c}-\frac {10 d \sqrt {a+b x}}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x}}{c x (c+d x)^{3/2}}\) |
-(Sqrt[a + b*x]/(c*x*(c + d*x)^(3/2))) + ((-10*d*Sqrt[a + b*x])/(3*c*(c + d*x)^(3/2)) + ((-2*d*(13*b*c - 15*a*d)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[ c + d*x]) - (6*(b*c - 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt [c + d*x])])/(Sqrt[a]*c^(3/2)))/(3*c))/(2*c)
3.6.96.3.1 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] && 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. \(652\) vs. \(2(120)=240\).
Time = 0.54 (sec) , antiderivative size = 653, normalized size of antiderivative = 4.41
| method | result | size |
| default | \(\frac {\left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{4} x^{3}-18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c \,d^{3} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d^{2} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{3} x^{2}-36 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{3} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c^{2} d^{2} x -18 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{3} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{4} x -30 a \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+26 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \,d^{2} x^{2}-40 a c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+36 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} d x -6 a \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}+6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{3}\right ) \sqrt {b x +a}}{6 c^{3} \left (a d -b c \right ) \sqrt {a c}\, x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(653\) |
1/6*(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^4*x^3-18*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^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)*b^2*c^2*d^2*x^3+30*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^3*x^2-36*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^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)*b^2*c^3*d*x^2+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^2*d^2*x-18*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^3*d*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^4*x-30*a*d^3*x^2* ((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+26*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2) *b*c*d^2*x^2-40*a*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+36*(a*c)^(1/ 2)*((b*x+a)*(d*x+c))^(1/2)*b*c^2*d*x-6*a*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a* c)^(1/2)+6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c^3)/c^3*(b*x+a)^(1/2)/(a *d-b*c)/(a*c)^(1/2)/x/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (120) = 240\).
Time = 0.58 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.53 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a b c^{4} - 3 \, a^{2} c^{3} d + {\left (13 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (9 \, a b c^{3} d - 10 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left ({\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x^{2} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x\right )}}, \frac {3 \, {\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (3 \, a b c^{4} - 3 \, a^{2} c^{3} d + {\left (13 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{2} + 2 \, {\left (9 \, a b c^{3} d - 10 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left ({\left (a b c^{5} d^{2} - a^{2} c^{4} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{6} d - a^{2} c^{5} d^{2}\right )} x^{2} + {\left (a b c^{7} - a^{2} c^{6} d\right )} x\right )}}\right ] \]
[-1/12*(3*((b^2*c^2*d^2 - 6*a*b*c*d^3 + 5*a^2*d^4)*x^3 + 2*(b^2*c^3*d - 6* a*b*c^2*d^2 + 5*a^2*c*d^3)*x^2 + (b^2*c^4 - 6*a*b*c^3*d + 5*a^2*c^2*d^2)*x )*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*(3*a*b*c^4 - 3*a^2*c^3*d + (13*a*b*c^2*d^2 - 15*a^2*c*d ^3)*x^2 + 2*(9*a*b*c^3*d - 10*a^2*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)) /((a*b*c^5*d^2 - a^2*c^4*d^3)*x^3 + 2*(a*b*c^6*d - a^2*c^5*d^2)*x^2 + (a*b *c^7 - a^2*c^6*d)*x), 1/6*(3*((b^2*c^2*d^2 - 6*a*b*c*d^3 + 5*a^2*d^4)*x^3 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^2 + (b^2*c^4 - 6*a*b*c^3*d + 5*a^2*c^2*d^2)*x)*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*(3*a*b*c^4 - 3*a^2*c^3*d + (13*a*b*c^2*d^2 - 15*a^2*c*d^3)*x^2 + 2*(9*a*b*c^3*d - 10*a^2*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a*b*c ^5*d^2 - a^2*c^4*d^3)*x^3 + 2*(a*b*c^6*d - a^2*c^5*d^2)*x^2 + (a*b*c^7 - a ^2*c^6*d)*x)]
\[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x}}{x^{2} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 571 vs. \(2 (120) = 240\).
Time = 1.12 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.86 \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{4} c^{4} d^{3} {\left | b \right |} - 6 \, a b^{3} c^{3} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}} + \frac {6 \, {\left (b^{5} c^{5} d^{2} {\left | b \right |} - 2 \, a b^{4} c^{4} d^{3} {\left | b \right |} + a^{2} b^{3} c^{3} d^{4} {\left | b \right |}\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (\sqrt {b d} b^{3} c - 5 \, \sqrt {b d} a b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} b^{5} c^{2} - 2 \, \sqrt {b d} a b^{4} c d + \sqrt {b d} a^{2} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{3} {\left | b \right |}} \]
-2/3*sqrt(b*x + a)*((5*b^4*c^4*d^3*abs(b) - 6*a*b^3*c^3*d^4*abs(b))*(b*x + a)/(b^3*c^7*d - a*b^2*c^6*d^2) + 6*(b^5*c^5*d^2*abs(b) - 2*a*b^4*c^4*d^3* abs(b) + a^2*b^3*c^3*d^4*abs(b))/(b^3*c^7*d - a*b^2*c^6*d^2))/(b^2*c + (b* x + a)*b*d - a*b*d)^(3/2) - (sqrt(b*d)*b^3*c - 5*sqrt(b*d)*a*b^2*d)*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)*b*c^3*abs(b)) - 2*(sqr t(b*d)*b^5*c^2 - 2*sqrt(b*d)*a*b^4*c*d + sqrt(b*d)*a^2*b^3*d^2 - sqrt(b*d) *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^3*c - 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^2*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*c^3*abs(b))
Timed out. \[ \int \frac {\sqrt {a+b x}}{x^2 (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \]