Integrand size = 22, antiderivative size = 191 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {3}{4} (b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} b \sqrt {a+b x} (c+d x)^{3/2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}} \]
-(b*x+a)^(3/2)*(d*x+c)^(3/2)/x-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^(1/2)+3/4*(a^2*d^2+6*a*b*c*d+b^2*c^2)*arct anh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(1/2)+3/2*b*(d* x+c)^(3/2)*(b*x+a)^(1/2)+3/4*(3*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)
Time = 0.69 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} (b x (5 c+2 d x)+a (-4 c+5 d x))}{x}+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}}\right ) \]
-3*Sqrt[a]*Sqrt[c]*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sq rt[a + b*x])] + ((Sqrt[a + b*x]*Sqrt[c + d*x]*(b*x*(5*c + 2*d*x) + a*(-4*c + 5*d*x)))/x + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(Sqrt[b]*Sqrt[d]))/4
Time = 0.34 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {108, 27, 171, 27, 171, 27, 175, 66, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \int \frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c+a d+2 b d x)}{2 x}dx-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \int \frac {\sqrt {a+b x} \sqrt {c+d x} (b c+a d+2 b d x)}{x}dx-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{2} \left (\frac {\int \frac {d \sqrt {c+d x} (2 a (b c+a d)+b (b c+3 a d) x)}{x \sqrt {a+b x}}dx}{2 d}+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \int \frac {\sqrt {c+d x} (2 a (b c+a d)+b (b c+3 a d) x)}{x \sqrt {a+b x}}dx+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {b \left (4 a c (b c+a d)+\left (b^2 c^2+6 a b d c+a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {4 a c (b c+a d)+\left (b^2 c^2+6 a b d c+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\left (a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+4 a c (a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+4 a c (a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+8 a c (a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {2 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}-8 \sqrt {a} \sqrt {c} (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )+\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)\right )+b \sqrt {a+b x} (c+d x)^{3/2}\right )-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}\) |
-(((a + b*x)^(3/2)*(c + d*x)^(3/2))/x) + (3*(b*Sqrt[a + b*x]*(c + d*x)^(3/ 2) + ((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x] + (-8*Sqrt[a]*Sqrt[c]*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + (2*(b^2 *c^2 + 6*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[ c + d*x])])/(Sqrt[b]*Sqrt[d]))/2)/2))/2
3.7.11.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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. \(418\) vs. \(2(149)=298\).
Time = 0.54 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.19
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} d^{2} x +18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b c d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2} x -12 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d x -12 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} x +4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d \,x^{2}+10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {b d}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\) | \(419\) |
1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/ 2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*a^2*d^2*x+18*ln(1/2*(2*b* d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2 )*a*b*c*d*x+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b* c)/(b*d)^(1/2))*(a*c)^(1/2)*b^2*c^2*x-12*(b*d)^(1/2)*ln((a*d*x+b*c*x+2*(a* c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*c*d*x-12*(b*d)^(1/2)*ln((a* d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b*c^2*x+4*(b*d )^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*d*x^2+10*(b*d)^(1/2)*(a*c)^( 1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d*x+10*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d* x+c))^(1/2)*b*c*x-8*((b*x+a)*(d*x+c))^(1/2)*a*c*(b*d)^(1/2)*(a*c)^(1/2))/( (b*x+a)*(d*x+c))^(1/2)/x/(b*d)^(1/2)/(a*c)^(1/2)
Time = 0.89 (sec) , antiderivative size = 1073, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \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 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, -\frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 6 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {a c} x \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 ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}, \frac {24 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \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 ) + 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d x}, \frac {12 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {-a c} x \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 ) - 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 4 \, a b c d + 5 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d x}\right ] \]
[1/16*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*log(8*b^2*d^2*x^2 + b ^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 12*(b^2*c*d + a*b*d^2)*sqr t(a*c)*x*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*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b*d^2)*x)*sqrt (b*x + a)*sqrt(d*x + c))/(b*d*x), -1/8*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)* sqrt(-b*d)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqr t(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 6*(b^2*c*d + a*b*d^2)*sqrt(a*c)*x*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) - 2*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b *d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/16*(24*(b^2*c*d + a*b*d^2 )*sqrt(-a*c)*x*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)) + 3*(b^2*c ^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b* c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/8*(12*(b^2*c*d + a* b*d^2)*sqrt(-a*c)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(...
\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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. 577 vs. \(2 (149) = 298\).
Time = 0.59 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.02 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b} + \frac {5 \, b c d^{2} {\left | b \right |} + 3 \, a d^{3} {\left | b \right |}}{b d^{2}}\right )} - \frac {3 \, {\left (b^{2} c^{2} {\left | b \right |} + 6 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d}} - \frac {24 \, {\left (\sqrt {b d} a b^{2} c^{2} {\left | b \right |} + \sqrt {b d} a^{2} b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} a b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{3} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b c d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}}}{8 \, b} \]
1/8*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d*ab s(b)/b + (5*b*c*d^2*abs(b) + 3*a*d^3*abs(b))/(b*d^2)) - 3*(b^2*c^2*abs(b) + 6*a*b*c*d*abs(b) + a^2*d^2*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b ^2*c + (b*x + a)*b*d - a*b*d))^2)/sqrt(b*d) - 24*(sqrt(b*d)*a*b^2*c^2*abs( b) + sqrt(b*d)*a^2*b*c*d*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*s qrt(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) - 16*(sqrt(b*d)*a*b^4*c^3*abs(b) - 2*sqrt(b*d)*a^2*b^3 *c^2*d*abs(b) + sqrt(b*d)*a^3*b^2*c*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*a*b^2*c^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*a^ 2*b*c*d*abs(b))/(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))/b
Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]