Integrand size = 20, antiderivative size = 520 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \] Output:
1/9*x*(a+b*ln(c*x^n))/d^(5/3)/(d^(1/3)+e^(1/3)*x)-(-1)^(1/3)*x*(a+b*ln(c*x ^n))/(1+(-1)^(1/3))^4/d^(5/3)/((-1)^(2/3)*d^(1/3)+e^(1/3)*x)+1/9*x*(a+b*ln (c*x^n))/d^(5/3)/(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)+(-1)^(1/3)*b*n*ln(-(-1)^(2 /3)*d^(1/3)-e^(1/3)*x)/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)-1/9*b*n*ln(d^(1/3) +e^(1/3)*x)/d^(5/3)/e^(1/3)+1/9*(-1)^(1/3)*b*n*ln(d^(1/3)+(-1)^(2/3)*e^(1/ 3)*x)/d^(5/3)/e^(1/3)+2/9*(a+b*ln(c*x^n))*ln(1+e^(1/3)*x/d^(1/3))/d^(5/3)/ e^(1/3)-2*I*3^(1/2)*(a+b*ln(c*x^n))*ln(1-(-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+ (-1)^(1/3))^5/d^(5/3)/e^(1/3)+2*(a+b*ln(c*x^n))*ln(1+(-1)^(2/3)*e^(1/3)*x/ d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)+2/9*b*n*polylog(2,-e^(1/3)*x/d^( 1/3))/d^(5/3)/e^(1/3)-2*I*3^(1/2)*b*n*polylog(2,(-1)^(1/3)*e^(1/3)*x/d^(1/ 3))/(1+(-1)^(1/3))^5/d^(5/3)/e^(1/3)+2*b*n*polylog(2,-(-1)^(2/3)*e^(1/3)*x /d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)
Time = 1.52 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {\frac {3 d^{2/3} x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^3}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\sqrt [3]{e}}+\frac {2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}+\frac {3 b n \left (\frac {\left (-1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt [3]{e} x \log (x)+\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\sqrt [3]{-1} \left (\frac {x \log (x)}{\sqrt [3]{d}+\sqrt [3]{e} x}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}\right )+\frac {-(-1)^{2/3} \sqrt [3]{e} x \log (x)+\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\frac {2 \sqrt [3]{-1} \left (\log (x) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\log (x) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (-1+\sqrt [3]{-1}\right ) \left (\log (x) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^2}}{9 d^{5/3}} \] Input:
Integrate[(a + b*Log[c*x^n])/(d + e*x^3)^2,x]
Output:
((3*d^(2/3)*x*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^3) - (2*Sqrt[3]*Ar cTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]]*(a - b*n*Log[x] + b*Log[c*x^n])) /e^(1/3) + (2*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d^(1/3) + e^(1/3)*x])/e^ (1/3) - ((a - b*n*Log[x] + b*Log[c*x^n])*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/e^(1/3) + (3*b*n*(((-1 + (-1)^(1/3))*((-1)^(1/3)*e^(1/3)*x* Log[x] + (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)*Log[-((-1)^(2/3)*d^(1/3)) - e^(1 /3)*x]))/((-1)^(2/3)*d^(1/3)*e^(1/3) + e^(2/3)*x) + (-1)^(1/3)*((x*Log[x]) /(d^(1/3) + e^(1/3)*x) - Log[d^(1/3) + e^(1/3)*x]/e^(1/3)) + (-((-1)^(2/3) *e^(1/3)*x*Log[x]) + (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)*Log[d^(1/3) + (-1)^( 2/3)*e^(1/3)*x])/(-((-1)^(1/3)*d^(1/3)*e^(1/3)) + e^(2/3)*x) + (2*(-1)^(1/ 3)*(Log[x]*Log[1 + (e^(1/3)*x)/d^(1/3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3) )]))/e^(1/3) - (2*(Log[x]*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)] + PolyLo g[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/e^(1/3) - (2*(-1 + (-1)^(1/3))*(Log [x]*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)] + PolyLog[2, -(((-1)^(2/3)*e^( 1/3)*x)/d^(1/3))]))/e^(1/3)))/(1 + (-1)^(1/3))^2)/(9*d^(5/3))
Time = 0.89 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2767, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \int \left (\frac {2 \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {2 (-1)^{5/6} \sqrt {3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}+\frac {2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {a+b \log \left (c x^n\right )}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac {(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )^2}+\frac {a+b \log \left (c x^n\right )}{\left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}\) |
Input:
Int[(a + b*Log[c*x^n])/(d + e*x^3)^2,x]
Output:
(x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*( a + b*Log[c*x^n]))/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/ 3)*x)) + (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x )) + ((-1)^(1/3)*b*n*Log[-((-1)^(2/3)*d^(1/3)) - e^(1/3)*x])/((1 + (-1)^(1 /3))^4*d^(5/3)*e^(1/3)) - (b*n*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3 )) + ((-1)^(1/3)*b*n*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(9*d^(5/3)*e^(1/ 3)) + (2*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/ 3)) - ((2*I)*Sqrt[3]*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^( 1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - ((2*I)*Sq rt[3]*b*n*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5* d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/( (1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x ^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.20 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {b x \ln \left (x^{n}\right )}{3 d \left (e \,x^{3}+d \right )}-\frac {2 b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right ) n \ln \left (x \right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right ) \ln \left (x^{n}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right ) n \ln \left (x \right )}{9 d e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right ) \ln \left (x^{n}\right )}{9 d e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {2 b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) n \ln \left (x \right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) \ln \left (x^{n}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}-\frac {b n \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {b n \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{18 d e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {b n \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b n \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}}\right )}{9 e d}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{3 d \left (e \,x^{3}+d \right )}+\frac {\frac {2 \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}}{d}\right )\) | \(598\) |
Input:
int((a+b*ln(c*x^n))/(e*x^3+d)^2,x,method=_RETURNVERBOSE)
Output:
1/3*b*x/d/(e*x^3+d)*ln(x^n)-2/9*b/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))/d*n*ln(x )+2/9*b/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))/d*ln(x^n)+1/9*b/d/e/(d/e)^(2/3)*ln (x^2-(d/e)^(1/3)*x+(d/e)^(2/3))*n*ln(x)-1/9*b/d/e/(d/e)^(2/3)*ln(x^2-(d/e) ^(1/3)*x+(d/e)^(2/3))*ln(x^n)-2/9*b/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/ 2)*(2/(d/e)^(1/3)*x-1))/d*n*ln(x)+2/9*b/e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3 ^(1/2)*(2/(d/e)^(1/3)*x-1))/d*ln(x^n)-1/9*b*n/e/(d/e)^(2/3)*ln(x+(d/e)^(1/ 3))/d+1/18*b*n/d/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))-1/9*b*n/e /(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))/d+2/9*b*n/e/d *sum(1/_R1^2*(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1)),_R1=RootOf(_Z^3*e+ d))+(1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csgn(I* c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3+1/2*I*Pi*b*csgn(I*c*x^n)^2*csg n(I*c)+b*ln(c)+a)*(1/3*x/d/(e*x^3+d)+2/3/d*(1/3/e/(d/e)^(2/3)*ln(x+(d/e)^( 1/3))-1/6/e/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3)*x+(d/e)^(2/3))+1/3/e/(d/e)^(2/3 )*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))))
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(e*x^3+d)^2,x, algorithm="fricas")
Output:
integral((b*log(c*x^n) + a)/(e^2*x^6 + 2*d*e*x^3 + d^2), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{3}\right )^{2}}\, dx \] Input:
integrate((a+b*ln(c*x**n))/(e*x**3+d)**2,x)
Output:
Integral((a + b*log(c*x**n))/(d + e*x**3)**2, x)
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*x^n))/(e*x^3+d)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(e*x^3+d)^2,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/(e*x^3 + d)^2, x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^3+d\right )}^2} \,d x \] Input:
int((a + b*log(c*x^n))/(d + e*x^3)^2,x)
Output:
int((a + b*log(c*x^n))/(d + e*x^3)^2, x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {-2 d^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a -2 d^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) a e \,x^{3}-d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a -d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {2}{3}}-e^{\frac {1}{3}} d^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right ) a e \,x^{3}+2 d^{\frac {4}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a +2 d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) a e \,x^{3}+9 e^{\frac {1}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) b \,d^{3}+9 e^{\frac {4}{3}} \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{6}+2 d e \,x^{3}+d^{2}}d x \right ) b \,d^{2} x^{3}+3 e^{\frac {1}{3}} a d x}{9 e^{\frac {1}{3}} d^{2} \left (e \,x^{3}+d \right )} \] Input:
int((a+b*log(c*x^n))/(e*x^3+d)^2,x)
Output:
( - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))* a*d - 2*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)) )*a*e*x**3 - d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)* a*d - d**(1/3)*log(d**(2/3) - e**(1/3)*d**(1/3)*x + e**(2/3)*x**2)*a*e*x** 3 + 2*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*a*d + 2*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*a*e*x**3 + 9*e**(1/3)*int(log(x**n*c)/(d**2 + 2*d*e*x**3 + e** 2*x**6),x)*b*d**3 + 9*e**(1/3)*int(log(x**n*c)/(d**2 + 2*d*e*x**3 + e**2*x **6),x)*b*d**2*e*x**3 + 3*e**(1/3)*a*d*x)/(9*e**(1/3)*d**2*(d + e*x**3))