Integrand size = 22, antiderivative size = 108 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {3 (b c-a d) \sqrt {a+b x}}{c^2 \sqrt {c+d x}}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {3 \sqrt {a} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \]
-3*(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*a^(1/2) /c^(5/2)-(b*x+a)^(3/2)/c/x/(d*x+c)^(1/2)+3*(-a*d+b*c)*(b*x+a)^(1/2)/c^2/(d *x+c)^(1/2)
Result contains complex when optimal does not.
Time = 10.89 (sec) , antiderivative size = 1648, normalized size of antiderivative = 15.26 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {\frac {\sqrt {c} (-2 b c x+a (c+3 d x)) \left (-4 a^2 d+b^2 x (3 c-d x)+b \sqrt {a-\frac {b c}{d}} \sqrt {a+b x} (-c+3 d x)+a \left (3 b c-5 b d x+4 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x}\right )\right )}{x \sqrt {c+d x} \left (b c \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )+b d x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )+a d \left (-4 \sqrt {a-\frac {b c}{d}}+4 \sqrt {a+b x}\right )\right )}+\frac {3 a b c \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 a^2 d^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 i \sqrt {a} b^2 c^2 \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {6 i a^{3/2} b c d \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 i a^{5/2} d^2 \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {3 a b c \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 a^2 d^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 i \sqrt {a} b^2 c^2 \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {6 i a^{3/2} b c d \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {3 i a^{5/2} d^2 \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}}{c^{5/2}} \]
((Sqrt[c]*(-2*b*c*x + a*(c + 3*d*x))*(-4*a^2*d + b^2*x*(3*c - d*x) + b*Sqr t[a - (b*c)/d]*Sqrt[a + b*x]*(-c + 3*d*x) + a*(3*b*c - 5*b*d*x + 4*Sqrt[a - (b*c)/d]*d*Sqrt[a + b*x])))/(x*Sqrt[c + d*x]*(b*c*(Sqrt[a - (b*c)/d] - 3 *Sqrt[a + b*x]) + b*d*x*(-3*Sqrt[a - (b*c)/d] + Sqrt[a + b*x]) + a*d*(-4*S qrt[a - (b*c)/d] + 4*Sqrt[a + b*x]))) + (3*a*b*c*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*S qrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqr t[a]*Sqrt[d]*Sqrt[b*c - a*d]] - (3*a^2*d^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sq rt[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[ d]*Sqrt[b*c - a*d]] + ((3*I)*Sqrt[a]*b^2*c^2*ArcTan[(Sqrt[b*c - 2*a*d - (2 *I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt [a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I )*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) - ((6*I)*a^(3/2)*b*c*d*ArcTan[(Sqrt[b* c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c] *Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + ((3*I)*a^(5/2)*d^2*Arc Tan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d* x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a* d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + (3*a*b*...
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {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.
| \(\displaystyle \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}}dx}{2 c}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {3 (b c-a d) \left (\frac {a \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}\right )}{2 c}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {3 (b c-a d) \left (\frac {2 a \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}\right )}{2 c}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3 (b c-a d) \left (\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}\right )}{2 c}-\frac {(a+b x)^{3/2}}{c x \sqrt {c+d x}}\) |
-((a + b*x)^(3/2)/(c*x*Sqrt[c + d*x])) + (3*(b*c - a*d)*((2*Sqrt[a + b*x]) /(c*Sqrt[c + d*x]) - (2*Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S qrt[c + d*x])])/c^(3/2)))/(2*c)
3.7.38.3.1 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. \(297\) vs. \(2(88)=176\).
Time = 0.55 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.76
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d x -3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{2 c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \sqrt {d x +c}}\) | \(298\) |
1/2*(b*x+a)^(1/2)*(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^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*c*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)*a^2*c*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)*a*b*c^2*x-6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d*x +4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c*x-2*((b*x+a)*(d*x+c))^(1/2)*a*c *(a*c)^(1/2))/c^2/((b*x+a)*(d*x+c))^(1/2)/x/(a*c)^(1/2)/(d*x+c)^(1/2)
Time = 0.38 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b c d - a d^{2}\right )} x^{2} + {\left (b c^{2} - a c d\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (a c - {\left (2 \, b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (c^{2} d x^{2} + c^{3} x\right )}}, \frac {3 \, {\left ({\left (b c d - a d^{2}\right )} x^{2} + {\left (b c^{2} - a c d\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (a c - {\left (2 \, b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c^{2} d x^{2} + c^{3} x\right )}}\right ] \]
[-1/4*(3*((b*c*d - a*d^2)*x^2 + (b*c^2 - a*c*d)*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^2 + (b*c^2 + a*c*d)*x)* sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4* (a*c - (2*b*c - 3*a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d*x^2 + c^3*x) , 1/2*(3*((b*c*d - a*d^2)*x^2 + (b*c^2 - a*c*d)*x)*sqrt(-a/c)*arctan(1/2*( 2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(a*c - (2*b*c - 3*a*d)*x)*sqrt(b*x + a)*s qrt(d*x + c))/(c^2*d*x^2 + c^3*x)]
\[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 475 vs. \(2 (88) = 176\).
Time = 1.02 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.40 \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} c^{2} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {b d} a b^{3} c - \sqrt {b d} a^{2} b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a b^{5} c^{2} - 2 \, \sqrt {b d} a^{2} b^{4} c d + \sqrt {b d} a^{3} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{2} {\left | b \right |}} \]
2*(b^3*c - a*b^2*d)*sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c^2 *abs(b)) - 3*(sqrt(b*d)*a*b^3*c - sqrt(b*d)*a^2*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^2*abs(b)) - 2*(sqrt(b*d)*a*b^5* c^2 - 2*sqrt(b*d)*a^2*b^4*c*d + sqrt(b*d)*a^3*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*a*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^2* 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^2*abs(b))
Timed out. \[ \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \]