Integrand size = 22, antiderivative size = 1198 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx =\text {Too large to display} \]
1/9*x*(a+b*ln(c*x^n))^3/d^(5/3)/(d^(1/3)+e^(1/3)*x)-(-1)^(1/3)*x*(a+b*ln(c *x^n))^3/(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))^3/d^(5/3)/(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)-1/3*b*n*(a+b*ln(c*x^ n))^2*ln(1+e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/9*(a+b*ln(c*x^n))^3*ln(1+e ^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+3*(-1)^(1/3)*b*n*(a+b*ln(c*x^n))^2*ln(1- (-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)+1/3*(-1)^(1 /3)*b*n*(a+b*ln(c*x^n))^2*ln(1+(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/ 3)-2/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/ 3)+2/3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3) +6*(-1)^(1/3)*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,(-1)^(1/3)*e^(1/3)*x/d^(1/ 3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)+2/3*(-1)^(1/3)*b^2*n^2*(a+b*ln(c*x^n) )*polylog(2,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+2/3*b^3*n^3*pol ylog(3,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-4/3*b^2*n^2*(a+b*ln(c*x^n))*pol ylog(3,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-6*(-1)^(1/3)*b^3*n^3*polylog(3, (-1)^(1/3)*e^(1/3)*x/d^(1/3))/(1+(-1)^(1/3))^4/d^(5/3)/e^(1/3)-2/3*(-1)^(1 /3)*b^3*n^3*polylog(3,-(-1)^(2/3)*e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)+4/3*b ^3*n^3*polylog(4,-e^(1/3)*x/d^(1/3))/d^(5/3)/e^(1/3)-4/9*(a+b*ln(c*x^n))^3 *ln(1-1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^(5/3)/e^(1/3)/(1-I*3^(1/2))-4 /3*b*n*(a+b*ln(c*x^n))^2*polylog(2,1/2*e^(1/3)*x*(1-I*3^(1/2))/d^(1/3))/d^ (5/3)/e^(1/3)/(1-I*3^(1/2))+8/3*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,1/2*e...
Time = 7.31 (sec) , antiderivative size = 2215, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\text {Result too large to show} \]
(x*(a + b*(-(n*Log[x]) + Log[c*x^n]))^3)/(3*d*(d + e*x^3)) + (2*ArcTan[(-d ^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))]*(a + b*(-(n*Log[x]) + Log[c*x^n]) )^3)/(3*Sqrt[3]*d^(5/3)*e^(1/3)) + (2*(a + b*(-(n*Log[x]) + Log[c*x^n]))^3 *Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) - ((a + b*(-(n*Log[x]) + Lo g[c*x^n]))^3*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(9*d^(5/3)*e^ (1/3)) + 3*b*n*(a + b*(-(n*Log[x]) + Log[c*x^n]))^2*(-1/3*((-1 + (-1)^(1/3 ))*((-((-1)^(1/3)/d^(1/3)) - ((-1)^(2/3)*d^(1/3) + e^(1/3)*x)^(-1))*Log[x] + ((-1)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - e^(1/3)*x])/d^(1/3)))/((1 + (-1 )^(1/3))^2*d^(4/3)*e^(1/3)) + ((-1)^(1/3)*((d^(-1/3) - (d^(1/3) + e^(1/3)* x)^(-1))*Log[x] - Log[d^(1/3) + e^(1/3)*x]/d^(1/3)))/(3*(1 + (-1)^(1/3))^2 *d^(4/3)*e^(1/3)) - (Log[x]/(e^(1/3)*((-1)^(1/3)*d^(1/3) - e^(1/3)*x)) - ( -(((-1)^(2/3)*Log[x])/d^(1/3)) + ((-1)^(2/3)*Log[d^(1/3) + (-1)^(2/3)*e^(1 /3)*x])/d^(1/3))/e^(1/3))/(3*(1 + (-1)^(1/3))^2*d^(4/3)) + (2*(-1)^(1/3)*( Log[x]*Log[1 + (e^(1/3)*x)/d^(1/3)] + PolyLog[2, -((e^(1/3)*x)/d^(1/3))])) /(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3)) - (2*(Log[x]*Log[1 - ((-1)^(1/3)*e ^(1/3)*x)/d^(1/3)] + PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*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)) ]))/(3*(1 + (-1)^(1/3))^2*d^(5/3)*e^(1/3))) + 3*b^2*n^2*(a + b*(-(n*Log[x] ) + Log[c*x^n]))*(((-1)^(1/3)*(Log[x]*((e^(1/3)*x*Log[x])/(d^(1/3) + e^...
Time = 1.79 (sec) , antiderivative size = 1198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \int \left (\frac {2 \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {2 (-1)^{5/6} \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}+\frac {2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac {(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 \sqrt [3]{-1} b^3 \operatorname {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b^3 \operatorname {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 \operatorname {PolyLog}\left (4,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {12 b^3 \operatorname {PolyLog}\left (4,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {6 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {12 b^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {3 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {6 b \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\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 )^3}{9 d^{5/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}\) |
(x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x *(a + b*Log[c*x^n])^3)/((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])^3)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^( 1/3)*x)) - (b*n*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(3*d^(5 /3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^3*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^ (5/3)*e^(1/3)) + (3*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^(1/3 )*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - ((2*I)*Sqrt[ 3]*(a + b*Log[c*x^n])^3*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1 )^(1/3))^5*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^ n])^3*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^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3) )])/(3*d^(5/3)*e^(1/3)) + (2*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((e^(1/3 )*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (6*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x ^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/ 3)*e^(1/3)) - ((6*I)*Sqrt[3]*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, ((-1)^(1/ 3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/ 3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3)) ])/(3*d^(5/3)*e^(1/3)) + (6*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(((-1)^(2 /3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b^3...
3.4.24.3.1 Defintions of rubi rules used
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]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{\left (e \,x^{3}+d \right )^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a ^3)/(e^2*x^6 + 2*d*e*x^3 + d^2), x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (e\,x^3+d\right )}^2} \,d x \]