Integrand size = 22, antiderivative size = 185 \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {b (3 b c-a d)}{a^2 c (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}-\frac {d \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {a+b x}}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2} c^{5/2}} \]
3*(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c ^(5/2)-b*(-a*d+3*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(1/2)-1/a/c/x /(b*x+a)^(1/2)/(d*x+c)^(1/2)-d*(3*a^2*d^2-2*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/ 2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(1/2)
Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-3 b^3 c^2 x (c+d x)-a^3 d^2 (c+3 d x)+a^2 b d \left (2 c^2+c d x-3 d^2 x^2\right )+a b^2 c \left (-c^2+c d x+2 d^2 x^2\right )}{a^2 c^2 (b c-a d)^2 x \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{5/2} c^{5/2}} \]
(-3*b^3*c^2*x*(c + d*x) - a^3*d^2*(c + 3*d*x) + a^2*b*d*(2*c^2 + c*d*x - 3 *d^2*x^2) + a*b^2*c*(-c^2 + c*d*x + 2*d^2*x^2))/(a^2*c^2*(b*c - a*d)^2*x*S qrt[a + b*x]*Sqrt[c + d*x]) + (3*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x ])/(Sqrt[c]*Sqrt[a + b*x])])/(a^(5/2)*c^(5/2))
Time = 0.33 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {114, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {3 (b c+a d)+4 b d x}{2 x (a+b x)^{3/2} (c+d x)^{3/2}}dx}{a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {3 (b c+a d)+4 b d x}{x (a+b x)^{3/2} (c+d x)^{3/2}}dx}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {2 \int \frac {3 (b c-a d) (b c+a d)+2 b d (3 b c-a d) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {3 (b c-a d) (b c+a d)+2 b d (3 b c-a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {2 d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {3 (b c-a d)^2 (b c+a d)}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 (b c-a d) (a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\frac {\frac {6 (b c-a d) (a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 d \sqrt {a+b x} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {6 (b c-a d) (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{a (b c-a d)}+\frac {2 b (3 b c-a d)}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {1}{a c x \sqrt {a+b x} \sqrt {c+d x}}\) |
-(1/(a*c*x*Sqrt[a + b*x]*Sqrt[c + d*x])) - ((2*b*(3*b*c - a*d))/(a*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + ((2*d*(3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d ^2)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (6*(b*c - a*d)*(b*c + a *d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^( 3/2)))/(a*(b*c - a*d)))/(2*a*c)
3.8.82.3.1 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[b*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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. \(896\) vs. \(2(161)=322\).
Time = 0.59 (sec) , antiderivative size = 897, normalized size of antiderivative = 4.85
| method | result | size |
| default | \(\frac {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^{3} b \,d^{4} 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 ) a^{2} b^{2} 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 ) a \,b^{3} c^{2} d^{2} 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^{4} c^{3} d \,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 ) a^{4} d^{4} 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^{2} b^{2} c^{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 ) b^{4} c^{4} 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^{4} c \,d^{3} 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^{3} b \,c^{2} 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 ) a^{2} b^{2} 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 ) a \,b^{3} c^{4} x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c \,d^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{2} d \,x^{2}-6 a^{3} d^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d x -6 b^{3} c^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-2 a^{3} c \,d^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,c^{2} d -2 a \,b^{2} c^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{2 c^{2} a^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \left (a d -b c \right )^{2} \sqrt {b x +a}\, \sqrt {d x +c}}\) | \(897\) |
1/2/c^2/a^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^3*b*d^4*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)*a^2*b^2*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)*a*b^3*c^2*d^2*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^4*c^3*d*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)*a^4*d^4*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^2*b^2*c^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)*b^4*c^4*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^4*c*d^3*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^3*b*c^2*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)*a^2*b ^2*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)*a*b^3*c^4*x-6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*d^3*x^2+4*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c*d^2*x^2-6*(a*c)^(1/2)*((b*x+a)*(d*x +c))^(1/2)*b^3*c^2*d*x^2-6*a^3*d^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2 *(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d^2*x+2*(a*c)^(1/2)*((b*x+a)* (d*x+c))^(1/2)*a*b^2*c^2*d*x-6*b^3*c^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/ 2)-2*a^3*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*(a*c)^(1/2)*((b*x+a)* (d*x+c))^(1/2)*a^2*b*c^2*d-2*a*b^2*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2) )/((b*x+a)*(d*x+c))^(1/2)/x/(a*c)^(1/2)/(a*d-b*c)^2/(b*x+a)^(1/2)/(d*x+...
Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (161) = 322\).
Time = 0.58 (sec) , antiderivative size = 940, normalized size of antiderivative = 5.08 \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x\right )}}, -\frac {3 \, {\left ({\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\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 (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3}\right )} x^{2} + {\left (3 \, a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + 3 \, a^{4} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{3} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{2} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x\right )}}\right ] \]
[1/4*(3*((b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^ 4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)*x^2 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^ 3*b*c^2*d^2 + a^4*c*d^3)*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*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (3*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4* b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(3*((b^4*c^3*d - a*b^3*c^2 *d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4 *d^4)*x^2 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x)*sqr t(-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*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3 *b*c*d^3)*x^2 + (3*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3 )*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^ 5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3* d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x)]
\[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 817 vs. \(2 (161) = 322\).
Time = 1.23 (sec) , antiderivative size = 817, normalized size of antiderivative = 4.42 \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b x + a} b^{2} d^{3}}{{\left (b^{2} c^{4} {\left | b \right |} - 2 \, a b c^{3} d {\left | b \right |} + a^{2} c^{2} d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{8} c^{4} - 8 \, \sqrt {b d} a b^{7} c^{3} d + 8 \, \sqrt {b d} a^{2} b^{6} c^{2} d^{2} - 4 \, \sqrt {b d} a^{3} b^{5} c d^{3} + \sqrt {b d} a^{4} b^{4} d^{4} - 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 )}^{2} b^{6} c^{3} - 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^{2} b^{4} c d^{2} + 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^{4} c^{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 )}^{4} a^{2} b^{2} d^{2}\right )}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{4} c^{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} a b^{3} c 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^{2} + 3 \, {\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^{2} c + {\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 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}\right )} {\left (a^{2} b c^{3} {\left | b \right |} - a^{3} c^{2} d {\left | b \right |}\right )}} + \frac {3 \, {\left (\sqrt {b d} b^{3} c + \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} a^{2} b c^{2} {\left | b \right |}} \]
-2*sqrt(b*x + a)*b^2*d^3/((b^2*c^4*abs(b) - 2*a*b*c^3*d*abs(b) + a^2*c^2*d ^2*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) - 2*(3*sqrt(b*d)*b^8*c^4 - 8*sqrt(b*d)*a*b^7*c^3*d + 8*sqrt(b*d)*a^2*b^6*c^2*d^2 - 4*sqrt(b*d)*a^3*b ^5*c*d^3 + sqrt(b*d)*a^4*b^4*d^4 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^3 - 2*sqrt(b*d)*(sqrt(b*d)*sq rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c*d^2 + 3*sqr t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b ^4*c^2 - 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^2*d^2)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3* b^3*d^3 - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) )^2*b^4*c^2 + 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* b*d))^2*a*b^3*c*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^2 + 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6)*(a^2*b*c^3*abs(b) - a^3*c^2*d*abs(b))) + 3*(sqrt(b*d) *b^3*c + 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))/(sqr t(-a*b*c*d)*a^2*b*c^2*abs(b))
Timed out. \[ \int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]