Integrand size = 20, antiderivative size = 149 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=-\frac {(2 b c-a d) \sqrt {c+d x}}{a^2 (a+b x)}-\frac {c \sqrt {c+d x}}{a x (a+b x)}+\frac {\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {\sqrt {b c-a d} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b}} \]
(-3*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))*c^(1/2)/a^3-(-a*d+4*b*c)*arc tanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/a^3/b^(1/2)- (-a*d+2*b*c)*(d*x+c)^(1/2)/a^2/(b*x+a)-c*(d*x+c)^(1/2)/a/x/(b*x+a)
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {\frac {a \sqrt {c+d x} (-a c-2 b c x+a d x)}{x (a+b x)}+\frac {\left (4 b^2 c^2-5 a b c d+a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}+\sqrt {c} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3} \]
((a*Sqrt[c + d*x]*(-(a*c) - 2*b*c*x + a*d*x))/(x*(a + b*x)) + ((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]]) /(Sqrt[b]*Sqrt[-(b*c) + a*d]) + Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d *x]/Sqrt[c]])/a^3
Time = 0.33 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {109, 27, 168, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int \frac {c (4 b c-3 a d)+d (3 b c-2 a d) x}{2 x (a+b x)^2 \sqrt {c+d x}}dx}{a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c (4 b c-3 a d)+d (3 b c-2 a d) x}{x (a+b x)^2 \sqrt {c+d x}}dx}{2 a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {\frac {\int \frac {c (4 b c-3 a d) (b c-a d)+d (2 b c-a d) x (b c-a d)}{x (a+b x) \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 \sqrt {c+d x} (2 b c-a d)}{a (a+b x)}}{2 a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {\frac {\frac {c (4 b c-3 a d) (b c-a d) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {(b c-a d)^2 (4 b c-a d) \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{a (b c-a d)}+\frac {2 \sqrt {c+d x} (2 b c-a d)}{a (a+b x)}}{2 a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {\frac {2 c (4 b c-3 a d) (b c-a d) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {2 (b c-a d)^2 (4 b c-a d) \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{a (b c-a d)}+\frac {2 \sqrt {c+d x} (2 b c-a d)}{a (a+b x)}}{2 a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {2 (b c-a d)^{3/2} (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a \sqrt {b}}-\frac {2 \sqrt {c} (4 b c-3 a d) (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}}{a (b c-a d)}+\frac {2 \sqrt {c+d x} (2 b c-a d)}{a (a+b x)}}{2 a}-\frac {c \sqrt {c+d x}}{a x (a+b x)}\) |
-((c*Sqrt[c + d*x])/(a*x*(a + b*x))) - ((2*(2*b*c - a*d)*Sqrt[c + d*x])/(a *(a + b*x)) + ((-2*Sqrt[c]*(4*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[Sqrt[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*c - a*d)))/(2*a)
3.5.67.3.1 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 + 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[(((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 = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(2 d^{3} \left (-\frac {c \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right ) \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(144\) |
| default | \(2 d^{3} \left (-\frac {c \left (\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a^{3} d^{3}}+\frac {\left (a d -b c \right ) \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\) | \(144\) |
| pseudoelliptic | \(\frac {x \left (a d -b c \right ) \left (a d -4 b c \right ) \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+4 \left (x \left (c^{\frac {3}{2}} b -\frac {3 a d \sqrt {c}}{4}\right ) \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-\frac {\left (2 b c x +a \left (-d x +c \right )\right ) a \sqrt {d x +c}}{4}\right ) \sqrt {\left (a d -b c \right ) b}}{x \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right ) a^{3}}\) | \(145\) |
| risch | \(-\frac {c \sqrt {d x +c}}{a^{2} x}-\frac {d \left (\frac {\frac {2 \left (-\frac {1}{2} a^{2} d^{2}+\frac {1}{2} a b c d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}-\frac {\left (a^{2} d^{2}-5 a b c d +4 b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a d}+\frac {\sqrt {c}\, \left (3 a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d}\right )}{a^{2}}\) | \(163\) |
2*d^3*(-c/a^3/d^3*(1/2*a*(d*x+c)^(1/2)/x+1/2*(3*a*d-4*b*c)/c^(1/2)*arctanh ((d*x+c)^(1/2)/c^(1/2)))+(a*d-b*c)/a^3/d^3*(1/2*a*d*(d*x+c)^(1/2)/((d*x+c) *b+a*d-b*c)+1/2*(a*d-4*b*c)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a *d-b*c)*b)^(1/2))))
Time = 0.29 (sec) , antiderivative size = 770, normalized size of antiderivative = 5.17 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\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^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} 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} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\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^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} 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} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\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} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {{\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\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^{2} c - 3 \, a b d\right )} x^{2} + {\left (4 \, a b c - 3 \, a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (a^{2} c + {\left (2 \, a b c - a^{2} d\right )} x\right )} \sqrt {d x + c}}{a^{3} b x^{2} + a^{4} x}\right ] \]
[-1/2*(((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*x)*sqrt((b*c - a*d)/b)*l og((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^2*c - 3*a*b*d)*x^2 + (4*a*b*c - 3*a^2*d)*x)*sqrt(c)*log((d*x - 2 *sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*c + (2*a*b*c - a^2*d)*x)*sqrt(d* x + c))/(a^3*b*x^2 + a^4*x), -1/2*(2*((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a ^2*d)*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^2*c - 3*a*b*d)*x^2 + (4*a*b*c - 3*a^2*d)*x)*sqrt(c)* log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*c + (2*a*b*c - a^2*d )*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x), -1/2*(2*((4*b^2*c - 3*a*b*d)*x^2 + (4*a*b*c - 3*a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + ((4*b ^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*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*c + (2*a*b*c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x), -(((4*b^2*c - a*b*d)*x^2 + (4*a*b*c - a^2*d)*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^2*c - 3*a*b*d)*x^2 + (4*a*b *c - 3*a^2*d)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (a^2*c + (2*a *b*c - a^2*d)*x)*sqrt(d*x + c))/(a^3*b*x^2 + a^4*x)]
\[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b c d - 2 \, \sqrt {d x + c} b c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} a d^{2} + 2 \, \sqrt {d x + c} a c d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \]
(4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b *d))/(sqrt(-b^2*c + a*b*d)*a^3) - (4*b*c^2 - 3*a*c*d)*arctan(sqrt(d*x + c) /sqrt(-c))/(a^3*sqrt(-c)) - (2*(d*x + c)^(3/2)*b*c*d - 2*sqrt(d*x + c)*b*c ^2*d - (d*x + c)^(3/2)*a*d^2 + 2*sqrt(d*x + c)*a*c*d^2)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)*a^2)
Time = 0.70 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.88 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^2} \, dx=-\frac {\frac {2\,\left (a\,c\,d^2-b\,c^2\,d\right )\,\sqrt {c+d\,x}}{a^2}-\frac {d\,\left (a\,d-2\,b\,c\right )\,{\left (c+d\,x\right )}^{3/2}}{a^2}}{\left (a\,d-2\,b\,c\right )\,\left (c+d\,x\right )+b\,{\left (c+d\,x\right )}^2+b\,c^2-a\,c\,d}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {6\,b\,\sqrt {c}\,d^7\,\sqrt {c+d\,x}}{6\,b\,c\,d^7-\frac {14\,b^2\,c^2\,d^6}{a}+\frac {8\,b^3\,c^3\,d^5}{a^2}}-\frac {14\,b^2\,c^{3/2}\,d^6\,\sqrt {c+d\,x}}{6\,a\,b\,c\,d^7-14\,b^2\,c^2\,d^6+\frac {8\,b^3\,c^3\,d^5}{a}}+\frac {8\,b^3\,c^{5/2}\,d^5\,\sqrt {c+d\,x}}{6\,a^2\,b\,c\,d^7-14\,a\,b^2\,c^2\,d^6+8\,b^3\,c^3\,d^5}\right )\,\left (3\,a\,d-4\,b\,c\right )}{a^3}-\frac {\mathrm {atanh}\left (\frac {2\,b\,c\,d^6\,\sqrt {b^2\,c-a\,b\,d}\,\sqrt {c+d\,x}}{2\,a\,b\,c\,d^7-10\,b^2\,c^2\,d^6+\frac {8\,b^3\,c^3\,d^5}{a}}-\frac {8\,b^2\,c^2\,d^5\,\sqrt {b^2\,c-a\,b\,d}\,\sqrt {c+d\,x}}{2\,a^2\,b\,c\,d^7-10\,a\,b^2\,c^2\,d^6+8\,b^3\,c^3\,d^5}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-4\,b\,c\right )}{a^3\,b} \]
- ((2*(a*c*d^2 - b*c^2*d)*(c + d*x)^(1/2))/a^2 - (d*(a*d - 2*b*c)*(c + d*x )^(3/2))/a^2)/((a*d - 2*b*c)*(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d) - (c^(1/2)*atanh((6*b*c^(1/2)*d^7*(c + d*x)^(1/2))/(6*b*c*d^7 - (14*b^2*c^2* d^6)/a + (8*b^3*c^3*d^5)/a^2) - (14*b^2*c^(3/2)*d^6*(c + d*x)^(1/2))/(6*a* b*c*d^7 - 14*b^2*c^2*d^6 + (8*b^3*c^3*d^5)/a) + (8*b^3*c^(5/2)*d^5*(c + d* x)^(1/2))/(8*b^3*c^3*d^5 - 14*a*b^2*c^2*d^6 + 6*a^2*b*c*d^7))*(3*a*d - 4*b *c))/a^3 - (atanh((2*b*c*d^6*(b^2*c - a*b*d)^(1/2)*(c + d*x)^(1/2))/(2*a*b *c*d^7 - 10*b^2*c^2*d^6 + (8*b^3*c^3*d^5)/a) - (8*b^2*c^2*d^5*(b^2*c - a*b *d)^(1/2)*(c + d*x)^(1/2))/(8*b^3*c^3*d^5 - 10*a*b^2*c^2*d^6 + 2*a^2*b*c*d ^7))*(-b*(a*d - b*c))^(1/2)*(a*d - 4*b*c))/(a^3*b)