Integrand size = 17, antiderivative size = 139 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \]
-1/2*ln(b*x+a)/b^(2/3)/(-a*d+b*c)^(1/3)+3/2*ln((-a*d+b*c)^(1/3)-b^(1/3)*(d *x+c)^(1/3))/b^(2/3)/(-a*d+b*c)^(1/3)+arctan(1/3*(1+2*b^(1/3)*(d*x+c)^(1/3 )/(-a*d+b*c)^(1/3))*3^(1/2))*3^(1/2)/b^(2/3)/(-a*d+b*c)^(1/3)
Time = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 b^{2/3} \sqrt [3]{-b c+a d}} \]
(-2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x)^(1/3))/(-(b*c) + a*d)^(1/3))/ Sqrt[3]] - 2*Log[(-(b*c) + a*d)^(1/3) + b^(1/3)*(c + d*x)^(1/3)] + Log[(-( b*c) + a*d)^(2/3) - b^(1/3)*(-(b*c) + a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3) *(c + d*x)^(2/3)])/(2*b^(2/3)*(-(b*c) + a*d)^(1/3))
Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{b}}-\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \int \frac {1}{\frac {(b c-a d)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{c+d x} \sqrt [3]{b c-a d}}{\sqrt [3]{b}}+(c+d x)^{2/3}}d\sqrt [3]{c+d x}}{2 b}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 \int \frac {1}{\frac {(b c-a d)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{c+d x} \sqrt [3]{b c-a d}}{\sqrt [3]{b}}+(c+d x)^{2/3}}d\sqrt [3]{c+d x}}{2 b}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}\) |
(Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(b*c - a*d)^(1/3))/Sqrt[3 ]])/(b^(2/3)*(b*c - a*d)^(1/3)) - Log[a + b*x]/(2*b^(2/3)*(b*c - a*d)^(1/3 )) + (3*Log[(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)])/(2*b^(2/3)*(b*c - a*d)^(1/3))
3.15.58.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 0.67 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (d x +c \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right )-2 \ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )+\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(143\) |
| derivativedivides | \(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(161\) |
| default | \(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(161\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(2*(d*x+c)^(1/3)-((a*d-b*c)/b)^(1/3))/(( a*d-b*c)/b)^(1/3))-2*ln((d*x+c)^(1/3)+((a*d-b*c)/b)^(1/3))+ln((d*x+c)^(2/3 )-((a*d-b*c)/b)^(1/3)*(d*x+c)^(1/3)+((a*d-b*c)/b)^(2/3)))/b/((a*d-b*c)/b)^ (1/3)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (108) = 216\).
Time = 0.24 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.10 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\left [\frac {\sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \log \left (\frac {2 \, b^{2} d x + 3 \, b^{2} c - a b d - \sqrt {3} {\left ({\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} - 3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}, \frac {2 \, \sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}}}{3 \, b}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}\right ] \]
[1/2*(sqrt(3)*(b^2*c - a*b*d)*sqrt(-(b^3*c - a*b^2*d)^(1/3)/(b*c - a*d))*l og((2*b^2*d*x + 3*b^2*c - a*b*d - sqrt(3)*((b^3*c - a*b^2*d)^(1/3)*(b*c - a*d) + (b^2*c - a*b*d)*(d*x + c)^(1/3) - 2*(b^3*c - a*b^2*d)^(2/3)*(d*x + c)^(2/3))*sqrt(-(b^3*c - a*b^2*d)^(1/3)/(b*c - a*d)) - 3*(b^3*c - a*b^2*d) ^(2/3)*(d*x + c)^(1/3))/(b*x + a)) - (b^3*c - a*b^2*d)^(2/3)*log((d*x + c) ^(2/3)*b^2 + (b^3*c - a*b^2*d)^(1/3)*(d*x + c)^(1/3)*b + (b^3*c - a*b^2*d) ^(2/3)) + 2*(b^3*c - a*b^2*d)^(2/3)*log((d*x + c)^(1/3)*b - (b^3*c - a*b^2 *d)^(1/3)))/(b^3*c - a*b^2*d), 1/2*(2*sqrt(3)*(b^2*c - a*b*d)*sqrt((b^3*c - a*b^2*d)^(1/3)/(b*c - a*d))*arctan(1/3*sqrt(3)*(2*(d*x + c)^(1/3)*b + (b ^3*c - a*b^2*d)^(1/3))*sqrt((b^3*c - a*b^2*d)^(1/3)/(b*c - a*d))/b) - (b^3 *c - a*b^2*d)^(2/3)*log((d*x + c)^(2/3)*b^2 + (b^3*c - a*b^2*d)^(1/3)*(d*x + c)^(1/3)*b + (b^3*c - a*b^2*d)^(2/3)) + 2*(b^3*c - a*b^2*d)^(2/3)*log(( d*x + c)^(1/3)*b - (b^3*c - a*b^2*d)^(1/3)))/(b^3*c - a*b^2*d)]
\[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt [3]{c + d x}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\text {Exception raised: ValueError} \]
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 = 0.34 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3} c - \sqrt {3} a b^{2} d} - \frac {\log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}} + \frac {\left (\frac {b c - a d}{b}\right )^{\frac {2}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c - a d} \]
3*(b^3*c - a*b^2*d)^(2/3)*arctan(1/3*sqrt(3)*(2*(d*x + c)^(1/3) + ((b*c - a*d)/b)^(1/3))/((b*c - a*d)/b)^(1/3))/(sqrt(3)*b^3*c - sqrt(3)*a*b^2*d) - 1/2*log((d*x + c)^(2/3) + (d*x + c)^(1/3)*((b*c - a*d)/b)^(1/3) + ((b*c - a*d)/b)^(2/3))/(b^3*c - a*b^2*d)^(1/3) + ((b*c - a*d)/b)^(2/3)*log(abs((d* x + c)^(1/3) - ((b*c - a*d)/b)^(1/3)))/(b*c - a*d)
Time = 0.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )}{b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}+\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}-\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}} \]
log(9*b*(c + d*x)^(1/3) - (9*b^3*c - 9*a*b^2*d)/(b^(4/3)*(b*c - a*d)^(2/3) ))/(b^(2/3)*(b*c - a*d)^(1/3)) + (log(9*b*(c + d*x)^(1/3) - ((3^(1/2)*1i - 1)^2*(9*b^3*c - 9*a*b^2*d))/(4*b^(4/3)*(b*c - a*d)^(2/3)))*(3^(1/2)*1i - 1))/(2*b^(2/3)*(b*c - a*d)^(1/3)) - (log(9*b*(c + d*x)^(1/3) - ((3^(1/2)*1 i + 1)^2*(9*b^3*c - 9*a*b^2*d))/(4*b^(4/3)*(b*c - a*d)^(2/3)))*(3^(1/2)*1i + 1))/(2*b^(2/3)*(b*c - a*d)^(1/3))
Time = 0.02 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{6}}}{b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{6}}}{b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}}}\right )-2 \,\mathrm {log}\left (\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )+\mathrm {log}\left (-b^{\frac {1}{6}} \left (d x +c \right )^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )+\mathrm {log}\left (b^{\frac {1}{6}} \left (d x +c \right )^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )}{2 b^{\frac {2}{3}} \left (a d -b c \right )^{\frac {1}{3}}} \]
( - 2*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6))) - 2*sqrt(3)*atan((b**(1/6)*(a* d - b*c)**(1/6)*sqrt(3) + 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b* c)**(1/6))) - 2*log((a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/3)) + log( - b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + (a*d - b*c)**(1/ 3) + b**(1/3)*(c + d*x)**(1/3)) + log(b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c )**(1/6)*sqrt(3) + (a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/3)))/(2*b** (2/3)*(a*d - b*c)**(1/3))