Integrand size = 22, antiderivative size = 142 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {5 c (b c-a d) \sqrt {c+d x}}{a^3 \sqrt {a+b x}}-\frac {5 (b c-a d) (c+d x)^{3/2}}{3 a^2 (a+b x)^{3/2}}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}+\frac {5 c^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2}} \]
-5/3*(-a*d+b*c)*(d*x+c)^(3/2)/a^2/(b*x+a)^(3/2)-(d*x+c)^(5/2)/a/x/(b*x+a)^ (3/2)+5*c^(3/2)*(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^( 1/2))/a^(7/2)-5*c*(-a*d+b*c)*(d*x+c)^(1/2)/a^3/(b*x+a)^(1/2)
Time = 10.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=-\frac {3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{3 a^{7/2} x (a+b x)^{3/2}} \]
-1/3*(3*a^(5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*( 4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt [a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(7/2)*x*(a + b*x)^(3/2))
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 105, 105, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}}dx}{2 a}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {5 (b c-a d) \left (\frac {c \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}}dx}{a}+\frac {2 (c+d x)^{3/2}}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {5 (b c-a d) \left (\frac {c \left (\frac {c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}\right )}{a}+\frac {2 (c+d x)^{3/2}}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5 (b c-a d) \left (\frac {c \left (\frac {2 c \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}\right )}{a}+\frac {2 (c+d x)^{3/2}}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {5 (b c-a d) \left (\frac {c \left (\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}\right )}{a}+\frac {2 (c+d x)^{3/2}}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {(c+d x)^{5/2}}{a x (a+b x)^{3/2}}\) |
-((c + d*x)^(5/2)/(a*x*(a + b*x)^(3/2))) - (5*(b*c - a*d)*((2*(c + d*x)^(3 /2))/(3*a*(a + b*x)^(3/2)) + (c*((2*Sqrt[c + d*x])/(a*Sqrt[a + b*x]) - (2* Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/a^(3/2)) )/a))/(2*a)
3.8.99.3.1 Defintions of rubi rules used
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[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -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. \(501\) vs. \(2(116)=232\).
Time = 1.71 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.54
| method | result | size |
| default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}-20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x +40 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 a^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}}}\) | \(502\) |
-1/6*(d*x+c)^(1/2)*(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*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)*b^3*c^3*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*b*c^2*d*x^2-30*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*b^2*c^3*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^3*c^2*d*x-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*b*c^3*x-4*(a*c)^(1/2)* ((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x^2-20*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2) *a*b*c*d*x^2+30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x^2-28*(a*c)^( 1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d*x+40*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 /2)*a*b*c^2*x+6*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/a^3/((b*x+a)* (d*x+c))^(1/2)/x/(a*c)^(1/2)/(b*x+a)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (116) = 232\).
Time = 0.44 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.56 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}, -\frac {15 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{3} + 2 \, {\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2} + {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, {\left (3 \, a^{2} c^{2} + {\left (15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (10 \, a b c^{2} - 7 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}}\right ] \]
[-1/12*(15*((b^3*c^2 - a*b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c*d)*x^2 + (a ^2*b*c^2 - a^3*c*d)*x)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a ^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)* sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(3*a^2*c^2 + (15*b^2*c^2 - 1 0*a*b*c*d - 2*a^2*d^2)*x^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b*x + a)*s qrt(d*x + c))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x), -1/6*(15*((b^3*c^2 - a* b^2*c*d)*x^3 + 2*(a*b^2*c^2 - a^2*b*c*d)*x^2 + (a^2*b*c^2 - a^3*c*d)*x)*sq rt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sq rt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 2*(3*a^2*c^2 + (15*b^2 *c^2 - 10*a*b*c*d - 2*a^2*d^2)*x^2 + 2*(10*a*b*c^2 - 7*a^2*c*d)*x)*sqrt(b* x + a)*sqrt(d*x + c))/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x)]
\[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 990 vs. \(2 (116) = 232\).
Time = 1.73 (sec) , antiderivative size = 990, normalized size of antiderivative = 6.97 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\frac {5 \, {\left (\sqrt {b d} b c^{3} {\left | b \right |} - \sqrt {b d} a c^{2} d {\left | b \right |}\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} a^{3} b} - \frac {2 \, {\left (\sqrt {b d} b^{3} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c^{3} d {\left | b \right |} + \sqrt {b d} a^{2} b c^{2} d^{2} {\left | b \right |} - \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 c^{3} {\left | b \right |} - \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 c^{2} d {\left | b \right |}\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 )} a^{3}} - \frac {4 \, {\left (6 \, \sqrt {b d} b^{7} c^{5} {\left | b \right |} - 23 \, \sqrt {b d} a b^{6} c^{4} d {\left | b \right |} + 32 \, \sqrt {b d} a^{2} b^{5} c^{3} d^{2} {\left | b \right |} - 18 \, \sqrt {b d} a^{3} b^{4} c^{2} d^{3} {\left | b \right |} + 2 \, \sqrt {b d} a^{4} b^{3} c d^{4} {\left | b \right |} + \sqrt {b d} a^{5} b^{2} d^{5} {\left | b \right |} - 12 \, \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^{5} c^{4} {\left | b \right |} + 36 \, \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^{4} c^{3} d {\left | b \right |} - 36 \, \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^{3} c^{2} d^{2} {\left | b \right |} + 12 \, \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^{2} c d^{3} {\left | b \right |} + 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 )}^{4} b^{3} c^{3} {\left | b \right |} - 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} a b^{2} c^{2} d {\left | b \right |} + 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} a^{3} d^{3} {\left | b \right |}\right )}}{3 \, {\left (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}\right )}^{3} a^{3} b^{2}} \]
5*(sqrt(b*d)*b*c^3*abs(b) - sqrt(b*d)*a*c^2*d*abs(b))*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^3*b) - 2*(sqrt(b*d)*b^3*c^4*abs(b) - 2*sqrt(b*d)*a*b^2*c^3*d*abs(b) + sqrt(b*d)*a^2*b*c^2*d^2*abs(b) - sqrt( b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b*c ^3*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^2*a*c^2*d*abs(b))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(s qrt(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)*a^3) - 4/3*(6*sqrt(b*d)*b^7*c^5*abs(b) - 23*sqrt(b*d)*a*b^6*c^4*d*abs(b) + 32*sqr t(b*d)*a^2*b^5*c^3*d^2*abs(b) - 18*sqrt(b*d)*a^3*b^4*c^2*d^3*abs(b) + 2*sq rt(b*d)*a^4*b^3*c*d^4*abs(b) + sqrt(b*d)*a^5*b^2*d^5*abs(b) - 12*sqrt(b*d) *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^5*c^4 *abs(b) + 36*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^4*c^3*d*abs(b) - 36*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^3*c^2*d^2*abs(b) + 12*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^2*c*d^3*abs(b) + 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b *x + a)*b*d - a*b*d))^4*b^3*c^3*abs(b) - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*...
Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,{\left (a+b\,x\right )}^{5/2}} \,d x \]