Integrand size = 22, antiderivative size = 268 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {b \left (15 b^2 c^2-2 a b c d-5 a^2 d^2\right )}{4 a^3 c^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {5 (b c+a d)}{4 a^2 c^2 x \sqrt {a+b x} \sqrt {c+d x}}+\frac {d (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x}}{4 a^3 c^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2} c^{7/2}} \]
-3/4*(5*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2) /(d*x+c)^(1/2))/a^(7/2)/c^(7/2)+1/4*b*(-5*a^2*d^2-2*a*b*c*d+15*b^2*c^2)/a^ 3/c^2/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(1/2)-1/2/a/c/x^2/(b*x+a)^(1/2)/(d* x+c)^(1/2)+5/4*(a*d+b*c)/a^2/c^2/x/(b*x+a)^(1/2)/(d*x+c)^(1/2)+1/4*d*(a*d+ b*c)*(15*a^2*d^2-22*a*b*c*d+15*b^2*c^2)*(b*x+a)^(1/2)/a^3/c^3/(-a*d+b*c)^2 /(d*x+c)^(1/2)
Time = 0.46 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {15 b^4 c^3 x^2 (c+d x)+a b^3 c^2 x \left (5 c^2-2 c d x-7 d^2 x^2\right )+a^4 d^2 \left (-2 c^2+5 c d x+15 d^2 x^2\right )-a^2 b^2 c \left (2 c^3+5 c^2 d x+10 c d^2 x^2+7 d^3 x^3\right )+a^3 b d \left (4 c^3-5 c^2 d x-2 c d^2 x^2+15 d^3 x^3\right )}{4 a^3 c^3 (b c-a d)^2 x^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{7/2} c^{7/2}} \]
(15*b^4*c^3*x^2*(c + d*x) + a*b^3*c^2*x*(5*c^2 - 2*c*d*x - 7*d^2*x^2) + a^ 4*d^2*(-2*c^2 + 5*c*d*x + 15*d^2*x^2) - a^2*b^2*c*(2*c^3 + 5*c^2*d*x + 10* c*d^2*x^2 + 7*d^3*x^3) + a^3*b*d*(4*c^3 - 5*c^2*d*x - 2*c*d^2*x^2 + 15*d^3 *x^3))/(4*a^3*c^3*(b*c - a*d)^2*x^2*Sqrt[a + b*x]*Sqrt[c + d*x]) - (3*(5*b ^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*S qrt[a + b*x])])/(4*a^(7/2)*c^(7/2))
Time = 0.44 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {114, 27, 168, 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.
| \(\displaystyle \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {5 (b c+a d)+6 b d x}{2 x^2 (a+b x)^{3/2} (c+d x)^{3/2}}dx}{2 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {5 (b c+a d)+6 b d x}{x^2 (a+b x)^{3/2} (c+d x)^{3/2}}dx}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )+20 b d (b c+a d) x}{2 x (a+b x)^{3/2} (c+d x)^{3/2}}dx}{a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )+20 b d (b c+a d) x}{x (a+b x)^{3/2} (c+d x)^{3/2}}dx}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {2 \int \frac {3 (b c-a d) \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )+2 b d \left (15 b^2 c^2-2 a b d c-5 a^2 d^2\right ) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {3 (b c-a d) \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )+2 b d \left (15 b^2 c^2-2 a b d c-5 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {2 d \sqrt {a+b x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {3 (b c-a d)^2 \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {3 (b c-a d) \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {6 (b c-a d) \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {\frac {2 d \sqrt {a+b x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {6 (b c-a d) \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{a (b c-a d)}+\frac {2 b \left (-5 a^2 d^2-2 a b c d+15 b^2 c^2\right )}{a \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {5 (a d+b c)}{a c x \sqrt {a+b x} \sqrt {c+d x}}}{4 a c}-\frac {1}{2 a c x^2 \sqrt {a+b x} \sqrt {c+d x}}\) |
-1/2*1/(a*c*x^2*Sqrt[a + b*x]*Sqrt[c + d*x]) - ((-5*(b*c + a*d))/(a*c*x*Sq rt[a + b*x]*Sqrt[c + d*x]) - ((2*b*(15*b^2*c^2 - 2*a*b*c*d - 5*a^2*d^2))/( a*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) + ((2*d*(b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (6*(b*c - a*d)*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*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))/(4*a*c)
3.8.83.3.1 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] && ILtQ[m, -1]
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. \(1371\) vs. \(2(230)=460\).
Time = 0.60 (sec) , antiderivative size = 1372, normalized size of antiderivative = 5.12
-1/8/c^3/a^3*(-30*a^4*d^4*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*b^4*c ^4*x^2*(a*c)^(1/2)*((b*x+a)*(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^5*d^5*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^5*c^5*x^3+14*a^2*b^2*c*d^3*x^3* (a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+14*a*b^3*c^2*d^2*x^3*(a*c)^(1/2)*((b*x +a)*(d*x+c))^(1/2)+4*a^3*b*c*d^3*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2 0*a^2*b^2*c^2*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*a*b^3*c^3*d*x^ 2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10*a^3*b*c^2*d^2*x*(a*c)^(1/2)*((b*x +a)*(d*x+c))^(1/2)+10*a^2*b^2*c^3*d*x*(a*c)^(1/2)*((b*x+a)*(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^4*b*d ^5*x^4+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^5*c^4*d*x^4+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^5*c*d^4*x^2+4*a^2*b^2*c^4*(a*c)^(1/2)*((b*x+a)*(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^4*c^5* x^2-10*a^4*c*d^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*a*b^3*c^4*x*(a*c )^(1/2)*((b*x+a)*(d*x+c))^(1/2)-8*a^3*b*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c) )^(1/2)-12*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^2*c*d^4*x^4-6*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^3*c^2*d^3*x^4-12*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^4*c^3*d^2*x^4+3*ln((a*d*x+b*c*x+2*(a*c)^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (230) = 460\).
Time = 1.58 (sec) , antiderivative size = 1244, normalized size of antiderivative = 4.64 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
[1/16*(3*((5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c *d^4 + 5*a^4*b*d^5)*x^4 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6 *a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b^3 *c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^2)*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*(2*a^3*b^2*c^5 - 4*a^4*b*c^4*d + 2*a^5*c^3*d^2 - (15*a*b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4)*x^3 - (15*a*b^4*c ^5 - 2*a^2*b^3*c^4*d - 10*a^3*b^2*c^3*d^2 - 2*a^4*b*c^2*d^3 + 15*a^5*c*d^4 )*x^2 - 5*(a^2*b^3*c^5 - a^3*b^2*c^4*d - a^4*b*c^3*d^2 + a^5*c^2*d^3)*x)*s qrt(b*x + a)*sqrt(d*x + c))/((a^4*b^3*c^6*d - 2*a^5*b^2*c^5*d^2 + a^6*b*c^ 4*d^3)*x^4 + (a^4*b^3*c^7 - a^5*b^2*c^6*d - a^6*b*c^5*d^2 + a^7*c^4*d^3)*x ^3 + (a^5*b^2*c^7 - 2*a^6*b*c^6*d + a^7*c^5*d^2)*x^2), 1/8*(3*((5*b^5*c^4* d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x ^4 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^ 4*b*c*d^4 + 5*a^5*d^5)*x^3 + (5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^ 3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^2)*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*(2*a^3*b^2*c^5 - 4*a^4*b*c^4*d + 2*a^5*c ^3*d^2 - (15*a*b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a...
\[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1204 vs. \(2 (230) = 460\).
Time = 3.37 (sec) , antiderivative size = 1204, normalized size of antiderivative = 4.49 \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
2*sqrt(b*x + a)*b^2*d^4/((b^2*c^5*abs(b) - 2*a*b*c^4*d*abs(b) + a^2*c^3*d^ 2*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) + 4*sqrt(b*d)*b^5/((a^3*b*c *abs(b) - a^4*d*abs(b))*(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)) - 3/4*(5*sqrt(b*d)*b^4*c^2 + 6*sqrt(b*d )*a*b^3*c*d + 5*sqrt(b*d)*a^2*b^2*d^2)*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*c^3*abs(b)) + 1/2*(7*sqrt(b*d)*b^10*c^5 - 21* sqrt(b*d)*a*b^9*c^4*d + 14*sqrt(b*d)*a^2*b^8*c^3*d^2 + 14*sqrt(b*d)*a^3*b^ 7*c^2*d^3 - 21*sqrt(b*d)*a^4*b^6*c*d^4 + 7*sqrt(b*d)*a^5*b^5*d^5 - 21*sqrt (b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^ 8*c^4 - 4*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^7*c^3*d + 50*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^6*c^2*d^2 - 4*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^3*b^5*c*d^3 - 21*sq rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2* a^4*b^4*d^4 + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c^3 + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^5*c^2*d + 35*sqrt(b*d)*(sqrt(b*d)*sq rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^4*c*d^2 + 21*sq rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))...
Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]