Integrand size = 22, antiderivative size = 204 \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=-\frac {d (23 b c-35 a d) \sqrt {a+b x}}{12 c^3 (c+d x)^{3/2}}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}-\frac {(5 b c-7 a d) \sqrt {a+b x}}{4 c^2 x (c+d x)^{3/2}}-\frac {5 d (11 b c-21 a d) \sqrt {a+b x}}{12 c^4 \sqrt {c+d x}}-\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} c^{9/2}} \]
-1/4*(35*a^2*d^2-30*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/ 2)/(d*x+c)^(1/2))/c^(9/2)/a^(1/2)-1/12*d*(-35*a*d+23*b*c)*(b*x+a)^(1/2)/c^ 3/(d*x+c)^(3/2)-1/2*a*(b*x+a)^(1/2)/c/x^2/(d*x+c)^(3/2)-1/4*(-7*a*d+5*b*c) *(b*x+a)^(1/2)/c^2/x/(d*x+c)^(3/2)-5/12*d*(-21*a*d+11*b*c)*(b*x+a)^(1/2)/c ^4/(d*x+c)^(1/2)
Time = 0.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-b c x \left (15 c^2+78 c d x+55 d^2 x^2\right )+a \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )}{12 c^4 x^2 (c+d x)^{3/2}}-\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} c^{9/2}} \]
(Sqrt[a + b*x]*(-(b*c*x*(15*c^2 + 78*c*d*x + 55*d^2*x^2)) + a*(-6*c^3 + 21 *c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3)))/(12*c^4*x^2*(c + d*x)^(3/2)) - ( (3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqr t[c]*Sqrt[a + b*x])])/(4*Sqrt[a]*c^(9/2))
Time = 0.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {109, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {a (5 b c-7 a d)+2 b (2 b c-3 a d) x}{2 x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (5 b c-7 a d)+2 b (2 b c-3 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {\int -\frac {a \left (3 b^2 c^2-30 a b d c+35 a^2 d^2-4 b d (5 b c-7 a d) x\right )}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 b^2 c^2-30 a b d c+35 a^2 d^2-4 b d (5 b c-7 a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {(b c-a d) \left (3 \left (3 b^2 c^2-30 a b d c+35 a^2 d^2\right )-2 b d (23 b c-35 a d) x\right )}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (3 b^2 c^2-30 a b d c+35 a^2 d^2\right )-2 b d (23 b c-35 a d) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 \int -\frac {3 (b c-a d) \left (3 b^2 c^2-30 a b d c+35 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}-\frac {10 d \sqrt {a+b x} (11 b c-21 a d)}{c \sqrt {c+d x}}}{3 c}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}-\frac {10 d \sqrt {a+b x} (11 b c-21 a d)}{c \sqrt {c+d x}}}{3 c}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\frac {\frac {\frac {6 \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {10 d \sqrt {a+b x} (11 b c-21 a d)}{c \sqrt {c+d x}}}{3 c}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {-\frac {6 \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {10 d \sqrt {a+b x} (11 b c-21 a d)}{c \sqrt {c+d x}}}{3 c}-\frac {2 d \sqrt {a+b x} (23 b c-35 a d)}{3 c (c+d x)^{3/2}}}{2 c}-\frac {\sqrt {a+b x} (5 b c-7 a d)}{c x (c+d x)^{3/2}}}{4 c}-\frac {a \sqrt {a+b x}}{2 c x^2 (c+d x)^{3/2}}\) |
-1/2*(a*Sqrt[a + b*x])/(c*x^2*(c + d*x)^(3/2)) + (-(((5*b*c - 7*a*d)*Sqrt[ a + b*x])/(c*x*(c + d*x)^(3/2))) + ((-2*d*(23*b*c - 35*a*d)*Sqrt[a + b*x]) /(3*c*(c + d*x)^(3/2)) + ((-10*d*(11*b*c - 21*a*d)*Sqrt[a + b*x])/(c*Sqrt[ c + d*x]) - (6*(3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt [a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3/2)))/(3*c))/(2*c))/(4*c )
3.7.46.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[(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. \(678\) vs. \(2(166)=332\).
Time = 0.56 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.33
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \left (105 \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^{4} x^{4}-90 \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^{3} x^{4}+9 \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^{2} c^{2} d^{2} x^{4}+210 \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^{3} x^{3}-180 \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} d^{2} x^{3}+18 \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^{2} c^{3} d \,x^{3}+105 \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^{2} d^{2} x^{2}-90 \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^{3} d \,x^{2}+9 \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^{2} c^{4} x^{2}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,d^{3} x^{3}+110 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \,d^{2} x^{3}-280 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \,d^{2} x^{2}+156 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} d \,x^{2}-42 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,c^{2} d x +30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{3} x +12 a \,c^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{24 c^{4} \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(679\) |
-1/24*(b*x+a)^(1/2)/c^4*(105*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^4*x^4-90*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^3*x^4+9*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^2*c^2*d^2*x^4+210*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^3*x^3-180*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*d^2*x^3+18*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^2*c^3*d*x^3+105 *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^2*d ^2*x^2-90*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^3*d*x^2+9*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^2*c^4*x^2-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d^3*x^3+110*(a *c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c*d^2*x^3-280*(a*c)^(1/2)*((b*x+a)*(d* x+c))^(1/2)*a*c*d^2*x^2+156*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c^2*d*x^ 2-42*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*c^2*d*x+30*(a*c)^(1/2)*((b*x+a) *(d*x+c))^(1/2)*b*c^3*x+12*a*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*c )^(1/2)/x^2/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)
Time = 1.57 (sec) , antiderivative size = 634, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{2} c^{4} + 5 \, {\left (11 \, a b c^{2} d^{2} - 21 \, a^{2} c d^{3}\right )} x^{3} + 2 \, {\left (39 \, a b c^{3} d - 70 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a c^{5} d^{2} x^{4} + 2 \, a c^{6} d x^{3} + a c^{7} x^{2}\right )}}\right ] \]
[1/48*(3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^3 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2 *c^2*d^2)*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*(6*a^2*c^4 + 5*(11*a*b*c^2*d^2 - 21*a^2*c* d^3)*x^3 + 2*(39*a*b*c^3*d - 70*a^2*c^2*d^2)*x^2 + 3*(5*a*b*c^4 - 7*a^2*c^ 3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^5*d^2*x^4 + 2*a*c^6*d*x^3 + a*c^ 7*x^2), 1/24*(3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4)*x^4 + 2*(3*b^ 2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^3 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2*c^2*d^2)*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*(6*a^2*c^4 + 5*(11*a*b*c^2*d^2 - 21*a^2*c*d^3)*x^3 + 2*(39*a*b *c^3*d - 70*a^2*c^2*d^2)*x^2 + 3*(5*a*b*c^4 - 7*a^2*c^3*d)*x)*sqrt(b*x + a )*sqrt(d*x + c))/(a*c^5*d^2*x^4 + 2*a*c^6*d*x^3 + a*c^7*x^2)]
\[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x^{3} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1234 vs. \(2 (166) = 332\).
Time = 1.45 (sec) , antiderivative size = 1234, normalized size of antiderivative = 6.05 \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
-2/3*sqrt(b*x + a)*((5*b^5*c^6*d^3*abs(b) - 14*a*b^4*c^5*d^4*abs(b) + 9*a^ 2*b^3*c^4*d^5*abs(b))*(b*x + a)/(b^3*c^9*d - a*b^2*c^8*d^2) + 3*(2*b^6*c^7 *d^2*abs(b) - 7*a*b^5*c^6*d^3*abs(b) + 8*a^2*b^4*c^5*d^4*abs(b) - 3*a^3*b^ 3*c^4*d^5*abs(b))/(b^3*c^9*d - a*b^2*c^8*d^2))/(b^2*c + (b*x + a)*b*d - a* b*d)^(3/2) - 1/4*(3*sqrt(b*d)*b^4*c^2 - 30*sqrt(b*d)*a*b^3*c*d + 35*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) - s qrt(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^4*abs(b)) - 1/2*(5*sqrt(b*d)*b^10*c^5 - 31*sqrt(b*d)*a*b^9*c^4*d + 74 *sqrt(b*d)*a^2*b^8*c^3*d^2 - 86*sqrt(b*d)*a^3*b^7*c^2*d^3 + 49*sqrt(b*d)*a ^4*b^6*c*d^4 - 11*sqrt(b*d)*a^5*b^5*d^5 - 15*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 + 52*sqrt(b*d)*(sqr t(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 - 26*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 - 44*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^5*c*d^3 + 33*sqrt(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 + 15*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 - 23*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 - 7*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^4*c*d^2 - 33*sqrt(b*d)*(sqrt(b*d)*sq...
Timed out. \[ \int \frac {(a+b x)^{3/2}}{x^3 (c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x^3\,{\left (c+d\,x\right )}^{5/2}} \,d x \]