Integrand size = 23, antiderivative size = 510 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac {e p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^2}+\frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{3 d^2} \] Output:
-3^(1/2)*b^(1/3)*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))/a^(1/ 3)/d-b^(1/3)*p*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/d-e*p*ln(-e*(a^(1/3)+b^(1/3)* x)/(b^(1/3)*d-a^(1/3)*e))*ln(e*x+d)/d^2-e*p*ln(-e*((-1)^(2/3)*a^(1/3)+b^(1 /3)*x)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))*ln(e*x+d)/d^2-e*p*ln((-1)^(1/3)*e *(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e))*ln(e*x+d )/d^2+1/2*b^(1/3)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)/d-ln (c*(b*x^3+a)^p)/d/x-1/3*e*ln(-b*x^3/a)*ln(c*(b*x^3+a)^p)/d^2+e*ln(e*x+d)*l n(c*(b*x^3+a)^p)/d^2-e*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-a^(1/3)*e))/ d^2-e*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e))/d^2-e* p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))/d^2-1/3*e*p* polylog(2,1+b*x^3/a)/d^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.17 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.77 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {9 b d p x^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )-2 a \left (3 e p x \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+3 e p x \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+3 e p x \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)+3 d \log \left (c \left (a+b x^3\right )^p\right )+e x \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-3 e x \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+3 e p x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+e p x \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )\right )}{6 a d^2 x} \] Input:
Integrate[Log[c*(a + b*x^3)^p]/(x^2*(d + e*x)),x]
Output:
(9*b*d*p*x^3*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)] - 2*a*(3*e*p*x*L og[(e*((-1)^(1/3)*a^(1/3) - b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e) ]*Log[d + e*x] + 3*e*p*x*Log[(e*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*d) + a^( 1/3)*e)]*Log[d + e*x] + 3*e*p*x*Log[(e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/( -(b^(1/3)*d) + (-1)^(2/3)*a^(1/3)*e)]*Log[d + e*x] + 3*d*Log[c*(a + b*x^3) ^p] + e*x*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p] - 3*e*x*Log[d + e*x]*Log[ c*(a + b*x^3)^p] + 3*e*p*x*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^( 1/3)*e)] + 3*e*p*x*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)* a^(1/3)*e)] + 3*e*p*x*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/ 3)*a^(1/3)*e)] + e*p*x*PolyLog[2, 1 + (b*x^3)/a]))/(6*a*d^2*x)
Time = 1.47 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2916, 2009}
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.
| \(\displaystyle \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (\frac {e^2 \log \left (c \left (a+b x^3\right )^p\right )}{d^2 (d+e x)}-\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac {\sqrt {3} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {e \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac {e p \operatorname {PolyLog}\left (2,\frac {b x^3}{a}+1\right )}{3 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}\) |
Input:
Int[Log[c*(a + b*x^3)^p]/(x^2*(d + e*x)),x]
Output:
-((Sqrt[3]*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(a ^(1/3)*d)) - (b^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/(a^(1/3)*d) - (e*p*Log[- ((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d^2 - ( e*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^( 1/3)*e))]*Log[d + e*x])/d^2 - (e*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3) *b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/d^2 + (b^(1 /3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)*d) - Log[ c*(a + b*x^3)^p]/(d*x) - (e*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d^2 ) + (e*Log[d + e*x]*Log[c*(a + b*x^3)^p])/d^2 - (e*p*PolyLog[2, (b^(1/3)*( d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e* x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, 1 + (b *x^3)/a])/(3*d^2)
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log [c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g , n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.56 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.57
| method | result | size |
| parts | \(\frac {e \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{2}}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-3 p b \left (\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 \textit {\_Z}^{2} b d +3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{3 d^{2} b}+\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{3 d^{2} b}\right )\) | \(292\) |
| risch | \(\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 \textit {\_Z}^{2} b d +3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{d^{2}}-\frac {p \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{d \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {p e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\) | \(415\) |
Input:
int(ln(c*(b*x^3+a)^p)/x^2/(e*x+d),x,method=_RETURNVERBOSE)
Output:
e*ln(e*x+d)*ln(c*(b*x^3+a)^p)/d^2-ln(c*(b*x^3+a)^p)/d/x-ln(c*(b*x^3+a)^p)* e/d^2*ln(x)-3*p*b*(1/3*e/d^2/b*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((- e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*d+3*_Z*b*d^2+a*e^3-b*d^3))+1/3/ d/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))-1/6/d/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a) ^(1/3)*x+(1/b*a)^(2/3))-1/3/d*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*( 2/(1/b*a)^(1/3)*x-1))-1/3*e/d^2/b*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/ _R1),_R1=RootOf(_Z^3*b+a)))
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^2/(e*x+d),x, algorithm="fricas")
Output:
integral(log((b*x^3 + a)^p*c)/(e*x^3 + d*x^2), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(ln(c*(b*x**3+a)**p)/x**2/(e*x+d),x)
Output:
Timed out
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^2/(e*x+d),x, algorithm="maxima")
Output:
integrate(log((b*x^3 + a)^p*c)/((e*x + d)*x^2), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^2/(e*x+d),x, algorithm="giac")
Output:
integrate(log((b*x^3 + a)^p*c)/((e*x + d)*x^2), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \] Input:
int(log(c*(a + b*x^3)^p)/(x^2*(d + e*x)),x)
Output:
int(log(c*(a + b*x^3)^p)/(x^2*(d + e*x)), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b p x -2 b^{\frac {2}{3}} a^{\frac {1}{3}} \left (\int \frac {\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right )}{e \,x^{2}+d x}d x \right ) e x -2 b^{\frac {2}{3}} a^{\frac {1}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right )-3 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b p x +\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) b x}{2 b^{\frac {2}{3}} a^{\frac {1}{3}} d x} \] Input:
int(log(c*(b*x^3+a)^p)/x^2/(e*x+d),x)
Output:
( - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*p*x - 2 *b**(2/3)*a**(1/3)*int(log((a + b*x**3)**p*c)/(d*x + e*x**2),x)*e*x - 2*b* *(2/3)*a**(1/3)*log((a + b*x**3)**p*c) - 3*log(a**(1/3) + b**(1/3)*x)*b*p* x + log((a + b*x**3)**p*c)*b*x)/(2*b**(2/3)*a**(1/3)*d*x)