Integrand size = 20, antiderivative size = 147 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=-\frac {c^2 \sqrt {c+d x}}{a^2 x}-\frac {(b c-a d)^2 \sqrt {c+d x}}{a^2 b (a+b x)}+\frac {c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {(b c-a d)^{3/2} (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 b^{3/2}} \]
c^(3/2)*(-5*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3-(-a*d+b*c)^(3/2) *(a*d+4*b*c)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/b^(3/2)-c ^2*(d*x+c)^(1/2)/a^2/x-(-a*d+b*c)^2*(d*x+c)^(1/2)/a^2/b/(b*x+a)
Time = 0.52 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a \sqrt {c+d x} \left (2 b^2 c^2 x+a^2 d^2 x+a b c (c-2 d x)\right )}{b x (a+b x)}+\frac {(-b c+a d)^{3/2} (4 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3} \]
(-((a*Sqrt[c + d*x]*(2*b^2*c^2*x + a^2*d^2*x + a*b*c*(c - 2*d*x)))/(b*x*(a + b*x))) + ((-(b*c) + a*d)^(3/2)*(4*b*c + a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d *x])/Sqrt[-(b*c) + a*d]])/b^(3/2) + c^(3/2)*(4*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^3
Time = 0.35 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {109, 27, 166, 25, 174, 73, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (4 b c-5 a d)+d (b c-2 a d) x)}{2 x (a+b x)^2}dx}{a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (4 b c-5 a d)+d (b c-2 a d) x)}{x (a+b x)^2}dx}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {\frac {2 \sqrt {c+d x} \left (\frac {2 b c^2}{a}+\frac {a d^2}{b}-3 c d\right )}{a+b x}-\frac {\int -\frac {b (4 b c-5 a d) c^2+d \left (2 b^2 c^2-2 a b d c-a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}}dx}{a b}}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {b (4 b c-5 a d) c^2+d \left (2 b^2 c^2-2 a b d c-a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}}dx}{a b}+\frac {2 \sqrt {c+d x} \left (\frac {2 b c^2}{a}+\frac {a d^2}{b}-3 c d\right )}{a+b x}}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {b c^2 (4 b c-5 a d) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {(b c-a d)^2 (a d+4 b c) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{a b}+\frac {2 \sqrt {c+d x} \left (\frac {2 b c^2}{a}+\frac {a d^2}{b}-3 c d\right )}{a+b x}}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {2 b c^2 (4 b c-5 a d) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 (b c-a d)^2 (a d+4 b c) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{a b}+\frac {2 \sqrt {c+d x} \left (\frac {2 b c^2}{a}+\frac {a d^2}{b}-3 c d\right )}{a+b x}}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 (b c-a d)^{3/2} (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 b c^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}}{a b}+\frac {2 \sqrt {c+d x} \left (\frac {2 b c^2}{a}+\frac {a d^2}{b}-3 c d\right )}{a+b x}}{2 a}-\frac {c (c+d x)^{3/2}}{a x (a+b x)}\) |
-((c*(c + d*x)^(3/2))/(a*x*(a + b*x))) - ((2*((2*b*c^2)/a - 3*c*d + (a*d^2 )/b)*Sqrt[c + d*x])/(a + b*x) + ((-2*b*c^(3/2)*(4*b*c - 5*a*d)*ArcTanh[Sqr t[c + d*x]/Sqrt[c]])/a + (2*(b*c - a*d)^(3/2)*(4*b*c + a*d)*ArcTanh[(Sqrt[ b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[b]))/(a*b))/(2*a)
3.5.73.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]
Time = 0.68 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(2 d^{3} \left (-\frac {c^{2} \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right )^{2} \left (-\frac {a d \sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a d +4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(154\) |
| default | \(2 d^{3} \left (-\frac {c^{2} \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right )^{2} \left (-\frac {a d \sqrt {d x +c}}{2 b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a d +4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(154\) |
| pseudoelliptic | \(\frac {x \left (a d +4 b c \right ) \left (a d -b c \right )^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+4 \sqrt {\left (a d -b c \right ) b}\, \left (x \left (c^{\frac {5}{2}} b -\frac {5 a d \,c^{\frac {3}{2}}}{4}\right ) b \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-\frac {\sqrt {d x +c}\, \left (2 b^{2} c^{2} x +a c \left (-2 d x +c \right ) b +a^{2} d^{2} x \right ) a}{4}\right )}{x \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right ) a^{3} b}\) | \(165\) |
| risch | \(-\frac {c^{2} \sqrt {d x +c}}{a^{2} x}+\frac {d \left (\frac {-\frac {a d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d x +c}}{b \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (a^{3} d^{3}+2 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b \sqrt {\left (a d -b c \right ) b}}}{a d}-\frac {c^{\frac {3}{2}} \left (5 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d}\right )}{a^{2}}\) | \(194\) |
2*d^3*(-c^2/a^3/d^3*(1/2*a*(d*x+c)^(1/2)/x+1/2*(5*a*d-4*b*c)/c^(1/2)*arcta nh((d*x+c)^(1/2)/c^(1/2)))+(a*d-b*c)^2/a^3/d^3*(-1/2*a*d/b*(d*x+c)^(1/2)/( (d*x+c)*b+a*d-b*c)+1/2*(a*d+4*b*c)/b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^ (1/2)/((a*d-b*c)*b)^(1/2))))
Time = 0.31 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.82 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (4 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 3 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left ({\left (4 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{2} + {\left (4 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt {d x + c}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \]
[-1/2*(((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b *c*d - a^3*d^2)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d *x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + ((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt( c) + 2*c)/x) + 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x)*sqr t(d*x + c))/(a^3*b^2*x^2 + a^4*b*x), -1/2*(2*((4*b^3*c^2 - 3*a*b^2*c*d - a ^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(-(b*c - a*d) /b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + ((4*b^3*c^ 2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(c)*log((d*x - 2 *sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c *d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a^4*b*x), -1/2*(2*((4*b^3*c ^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(-c)*arctan(sqr t(d*x + c)*sqrt(-c)/c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a *b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b* c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x)*sqrt(d*x + c))/(a^3*b^2*x^2 + a^4*b*x), -(((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3* a^2*b*c*d - a^3*d^2)*x)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt( -(b*c - a*d)/b)/(b*c - a*d)) + ((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c ^2 - 5*a^2*b*c*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (a^2*b...
\[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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. 260 vs. \(2 (127) = 254\).
Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=-\frac {{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} + \frac {{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3} b} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{2} d - 2 \, \sqrt {d x + c} b^{2} c^{3} d - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{2} + 3 \, \sqrt {d x + c} a b c^{2} d^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{3} - \sqrt {d x + c} a^{2} c d^{3}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2} b} \]
-(4*b*c^3 - 5*a*c^2*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)) + (4* b^3*c^3 - 7*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*arctan(sqrt(d*x + c)*b/ sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3*b) - (2*(d*x + c)^(3/2)*b^ 2*c^2*d - 2*sqrt(d*x + c)*b^2*c^3*d - 2*(d*x + c)^(3/2)*a*b*c*d^2 + 3*sqrt (d*x + c)*a*b*c^2*d^2 + (d*x + c)^(3/2)*a^2*d^3 - sqrt(d*x + c)*a^2*c*d^3) /(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2*b)
Time = 0.39 (sec) , antiderivative size = 1127, normalized size of antiderivative = 7.67 \[ \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx=\frac {\frac {\sqrt {c+d\,x}\,\left (a^2\,c\,d^3-3\,a\,b\,c^2\,d^2+2\,b^2\,c^3\,d\right )}{a^2\,b}-\frac {d\,{\left (c+d\,x\right )}^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^2\,b}}{\left (a\,d-2\,b\,c\right )\,\left (c+d\,x\right )+b\,{\left (c+d\,x\right )}^2+b\,c^2-a\,c\,d}-\frac {\mathrm {atanh}\left (\frac {10\,d^9\,\sqrt {c^3}\,\sqrt {c+d\,x}}{10\,c^2\,d^9+\frac {32\,b\,c^3\,d^8}{a}-\frac {132\,b^2\,c^4\,d^7}{a^2}+\frac {130\,b^3\,c^5\,d^6}{a^3}-\frac {40\,b^4\,c^6\,d^5}{a^4}}+\frac {32\,c\,d^8\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,c^3\,d^8+\frac {10\,a\,c^2\,d^9}{b}-\frac {132\,b\,c^4\,d^7}{a}+\frac {130\,b^2\,c^5\,d^6}{a^2}-\frac {40\,b^3\,c^6\,d^5}{a^3}}-\frac {132\,b\,c^2\,d^7\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a\,c^3\,d^8-132\,b\,c^4\,d^7+\frac {130\,b^2\,c^5\,d^6}{a}+\frac {10\,a^2\,c^2\,d^9}{b}-\frac {40\,b^3\,c^6\,d^5}{a^2}}+\frac {130\,b^2\,c^3\,d^6\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a^2\,c^3\,d^8+130\,b^2\,c^5\,d^6-\frac {40\,b^3\,c^6\,d^5}{a}+\frac {10\,a^3\,c^2\,d^9}{b}-132\,a\,b\,c^4\,d^7}-\frac {40\,b^3\,c^4\,d^5\,\sqrt {c^3}\,\sqrt {c+d\,x}}{32\,a^3\,c^3\,d^8-40\,b^3\,c^6\,d^5+130\,a\,b^2\,c^5\,d^6-132\,a^2\,b\,c^4\,d^7+\frac {10\,a^4\,c^2\,d^9}{b}}\right )\,\left (5\,a\,d-4\,b\,c\right )\,\sqrt {c^3}}{a^3}-\frac {\mathrm {atanh}\left (\frac {30\,c^3\,d^6\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{14\,a^3\,c^2\,d^9+110\,b^3\,c^5\,d^6-82\,a\,b^2\,c^4\,d^7-4\,a^2\,b\,c^3\,d^8+\frac {2\,a^4\,c\,d^{10}}{b}-\frac {40\,b^4\,c^6\,d^5}{a}}-\frac {2\,c\,d^8\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,b^3\,c^3\,d^8-14\,a\,b^2\,c^2\,d^9+\frac {82\,b^4\,c^4\,d^7}{a}-\frac {110\,b^5\,c^5\,d^6}{a^2}+\frac {40\,b^6\,c^6\,d^5}{a^3}-2\,a^2\,b\,c\,d^{10}}+\frac {18\,c^2\,d^7\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{2\,a^3\,c\,d^{10}-82\,b^3\,c^4\,d^7-4\,a\,b^2\,c^3\,d^8+14\,a^2\,b\,c^2\,d^9+\frac {110\,b^4\,c^5\,d^6}{a}-\frac {40\,b^5\,c^6\,d^5}{a^2}}+\frac {40\,c^4\,d^5\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,a^3\,c^3\,d^8+40\,b^3\,c^6\,d^5-110\,a\,b^2\,c^5\,d^6+82\,a^2\,b\,c^4\,d^7-\frac {2\,a^5\,c\,d^{10}}{b^2}-\frac {14\,a^4\,c^2\,d^9}{b}}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (a\,d+4\,b\,c\right )}{a^3\,b^3} \]
(((c + d*x)^(1/2)*(a^2*c*d^3 + 2*b^2*c^3*d - 3*a*b*c^2*d^2))/(a^2*b) - (d* (c + d*x)^(3/2)*(a^2*d^2 + 2*b^2*c^2 - 2*a*b*c*d))/(a^2*b))/((a*d - 2*b*c) *(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d) - (atanh((10*d^9*(c^3)^(1/2)*( c + d*x)^(1/2))/(10*c^2*d^9 + (32*b*c^3*d^8)/a - (132*b^2*c^4*d^7)/a^2 + ( 130*b^3*c^5*d^6)/a^3 - (40*b^4*c^6*d^5)/a^4) + (32*c*d^8*(c^3)^(1/2)*(c + d*x)^(1/2))/(32*c^3*d^8 + (10*a*c^2*d^9)/b - (132*b*c^4*d^7)/a + (130*b^2* c^5*d^6)/a^2 - (40*b^3*c^6*d^5)/a^3) - (132*b*c^2*d^7*(c^3)^(1/2)*(c + d*x )^(1/2))/(32*a*c^3*d^8 - 132*b*c^4*d^7 + (130*b^2*c^5*d^6)/a + (10*a^2*c^2 *d^9)/b - (40*b^3*c^6*d^5)/a^2) + (130*b^2*c^3*d^6*(c^3)^(1/2)*(c + d*x)^( 1/2))/(32*a^2*c^3*d^8 + 130*b^2*c^5*d^6 - (40*b^3*c^6*d^5)/a + (10*a^3*c^2 *d^9)/b - 132*a*b*c^4*d^7) - (40*b^3*c^4*d^5*(c^3)^(1/2)*(c + d*x)^(1/2))/ (32*a^3*c^3*d^8 - 40*b^3*c^6*d^5 + 130*a*b^2*c^5*d^6 - 132*a^2*b*c^4*d^7 + (10*a^4*c^2*d^9)/b))*(5*a*d - 4*b*c)*(c^3)^(1/2))/a^3 - (atanh((30*c^3*d^ 6*(c + d*x)^(1/2)*(b^6*c^3 - a^3*b^3*d^3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d )^(1/2))/(14*a^3*c^2*d^9 + 110*b^3*c^5*d^6 - 82*a*b^2*c^4*d^7 - 4*a^2*b*c^ 3*d^8 + (2*a^4*c*d^10)/b - (40*b^4*c^6*d^5)/a) - (2*c*d^8*(c + d*x)^(1/2)* (b^6*c^3 - a^3*b^3*d^3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d)^(1/2))/(4*b^3*c^ 3*d^8 - 14*a*b^2*c^2*d^9 + (82*b^4*c^4*d^7)/a - (110*b^5*c^5*d^6)/a^2 + (4 0*b^6*c^6*d^5)/a^3 - 2*a^2*b*c*d^10) + (18*c^2*d^7*(c + d*x)^(1/2)*(b^6*c^ 3 - a^3*b^3*d^3 + 3*a^2*b^4*c*d^2 - 3*a*b^5*c^2*d)^(1/2))/(2*a^3*c*d^10...