Integrand size = 23, antiderivative size = 674 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{3 d^3} \] Output:
-1/2*3^(1/2)*b^(2/3)*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))/a ^(2/3)/d+3^(1/2)*b^(1/3)*e*p*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1 /3))/a^(1/3)/d^2+1/2*b^(2/3)*p*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/d+b^(1/3)*e*p *ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/d^2+e^2*p*ln(-e*(a^(1/3)+b^(1/3)*x)/(b^(1/3 )*d-a^(1/3)*e))*ln(e*x+d)/d^3+e^2*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^3+e^2*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^3- 1/4*b^(2/3)*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/d-1/2*b^(1 /3)*e*p*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)/d^2-1/2*ln(c*(b* x^3+a)^p)/d/x^2+e*ln(c*(b*x^3+a)^p)/d^2/x+1/3*e^2*ln(-b*x^3/a)*ln(c*(b*x^3 +a)^p)/d^3-e^2*ln(e*x+d)*ln(c*(b*x^3+a)^p)/d^3+e^2*p*polylog(2,b^(1/3)*(e* x+d)/(b^(1/3)*d-a^(1/3)*e))/d^3+e^2*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d +(-1)^(1/3)*a^(1/3)*e))/d^3+e^2*p*polylog(2,b^(1/3)*(e*x+d)/(b^(1/3)*d-(-1 )^(2/3)*a^(1/3)*e))/d^3+1/3*e^2*p*polylog(2,1+b*x^3/a)/d^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.50 (sec) , antiderivative size = 542, normalized size of antiderivative = 0.80 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {6 \sqrt {3} b^{2/3} d^2 p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {18 b d e p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )}{a}+\frac {6 b^{2/3} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)-\frac {3 b^{2/3} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac {6 d^2 \log \left (c \left (a+b x^3\right )^p\right )}{x^2}+\frac {12 d e \log \left (c \left (a+b x^3\right )^p\right )}{x}+4 e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-12 e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+4 e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{12 d^3} \] Input:
Integrate[Log[c*(a + b*x^3)^p]/(x^3*(d + e*x)),x]
Output:
((-6*Sqrt[3]*b^(2/3)*d^2*p*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^ (2/3) - (18*b*d*e*p*x^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*x^3)/a)])/a + (6*b^(2/3)*d^2*p*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + 12*e^2*p*Log[(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] + 12*e^2*p*Log[(e*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*d) + a^(1/3)*e)]*L og[d + e*x] + 12*e^2*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] - (3*b^(2/3)*d^2*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) - (6*d^2*Log[c*(a + b*x^3)^p])/ x^2 + (12*d*e*Log[c*(a + b*x^3)^p])/x + 4*e^2*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p] - 12*e^2*Log[d + e*x]*Log[c*(a + b*x^3)^p] + 12*e^2*p*PolyLog[2 , (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)] + 12*e^2*p*PolyLog[2, (b^(1 /3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)] + 12*e^2*p*PolyLog[2, ( b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)] + 4*e^2*p*PolyLog[2 , 1 + (b*x^3)/a])/(12*d^3)
Time = 1.80 (sec) , antiderivative size = 674, 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.
| \(\displaystyle \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (-\frac {e^3 \log \left (c \left (a+b x^3\right )^p\right )}{d^3 (d+e x)}+\frac {e^2 \log \left (c \left (a+b x^3\right )^p\right )}{d^3 x}-\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x^2}+\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b x^3}{a}+1\right )}{3 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^3}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}\) |
Input:
Int[Log[c*(a + b*x^3)^p]/(x^3*(d + e*x)),x]
Output:
-1/2*(Sqrt[3]*b^(2/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]) /(a^(2/3)*d) + (Sqrt[3]*b^(1/3)*e*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3 ]*a^(1/3))])/(a^(1/3)*d^2) + (b^(2/3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*a^(2/ 3)*d) + (b^(1/3)*e*p*Log[a^(1/3) + b^(1/3)*x])/(a^(1/3)*d^2) + (e^2*p*Log[ -((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d^3 + (e^2*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^3 + (e^2*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^3 - (b^(2/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*a^(2/3)*d) - (b^(1/3)*e*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)*d ^2) - Log[c*(a + b*x^3)^p]/(2*d*x^2) + (e*Log[c*(a + b*x^3)^p])/(d^2*x) + (e^2*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d^3) - (e^2*Log[d + e*x]*L og[c*(a + b*x^3)^p])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)* d - a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/ 3)*d - (-1)^(2/3)*a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, 1 + (b*x^3)/a])/(3* d^3)
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 = 2.40 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.62
| method | result | size |
| parts | \(-\frac {e^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{3}}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{2} x}-\frac {3 p b \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d b \left (\frac {a}{b}\right )^{\frac {2}{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 {2}{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 {2}{3}}}-\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 d^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 e \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^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 e^{2} \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^{3} b}-\frac {2 e^{2} \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^{3} b}\right )}{2}\) | \(421\) |
| risch | \(-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e}{d^{2} x}+\frac {p \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{2 d \left (\frac {a}{b}\right )^{\frac {2}{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 )}{4 d \left (\frac {a}{b}\right )^{\frac {2}{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 )}{2 d \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {p e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p \,e^{2} \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^{3}}+\frac {p \,e^{2} \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^{3}}+\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^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )\) | \(548\) |
Input:
int(ln(c*(b*x^3+a)^p)/x^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-e^2*ln(e*x+d)*ln(c*(b*x^3+a)^p)/d^3-1/2*ln(c*(b*x^3+a)^p)/d/x^2+ln(c*(b*x ^3+a)^p)*e^2/d^3*ln(x)+e*ln(c*(b*x^3+a)^p)/d^2/x-3/2*p*b*(-1/3/d/b/(1/b*a) ^(2/3)*ln(x+(1/b*a)^(1/3))+1/6/d/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1 /b*a)^(2/3))-1/3/d/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^( 1/3)*x-1))-2/3/d^2*e/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/3/d^2*e/b/(1/b* a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/3/d^2*e*3^(1/2)/b/(1/b*a) ^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+2/3*e^2/d^3/b*sum(ln(x)*l n((_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^3*b+a))-2/3*e^2/d^3/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)))
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^3/(e*x+d),x, algorithm="fricas")
Output:
integral(log((b*x^3 + a)^p*c)/(e*x^4 + d*x^3), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(ln(c*(b*x**3+a)**p)/x**3/(e*x+d),x)
Output:
Timed out
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^3/(e*x+d),x, algorithm="maxima")
Output:
integrate(log((b*x^3 + a)^p*c)/((e*x + d)*x^3), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate(log(c*(b*x^3+a)^p)/x^3/(e*x+d),x, algorithm="giac")
Output:
integrate(log((b*x^3 + a)^p*c)/((e*x + d)*x^3), x)
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \] Input:
int(log(c*(a + b*x^3)^p)/(x^3*(d + e*x)),x)
Output:
int(log(c*(a + b*x^3)^p)/(x^3*(d + e*x)), x)
\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {4 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b e p \,x^{2}-2 b^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) d p \,x^{2}+4 b^{\frac {2}{3}} a^{\frac {2}{3}} \left (\int \frac {\mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right )}{e \,x^{2}+d x}d x \right ) e^{2} x^{2}-2 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d +4 b^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) e x +6 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b e p \,x^{2}-2 a^{\frac {1}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) b e \,x^{2}+3 b^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) d p \,x^{2}-b^{\frac {4}{3}} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d \,x^{2}}{4 b^{\frac {2}{3}} a^{\frac {2}{3}} d^{2} x^{2}} \] Input:
int(log(c*(b*x^3+a)^p)/x^3/(e*x+d),x)
Output:
(4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b*e *p*x**2 - 2*b**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt (3)))*b*d*p*x**2 + 4*b**(2/3)*a**(2/3)*int(log((a + b*x**3)**p*c)/(d*x + e *x**2),x)*e**2*x**2 - 2*b**(2/3)*a**(2/3)*log((a + b*x**3)**p*c)*d + 4*b** (2/3)*a**(2/3)*log((a + b*x**3)**p*c)*e*x + 6*a**(1/3)*log(a**(1/3) + b**( 1/3)*x)*b*e*p*x**2 - 2*a**(1/3)*log((a + b*x**3)**p*c)*b*e*x**2 + 3*b**(1/ 3)*log(a**(1/3) + b**(1/3)*x)*b*d*p*x**2 - b**(1/3)*log((a + b*x**3)**p*c) *b*d*x**2)/(4*b**(2/3)*a**(2/3)*d**2*x**2)