Integrand size = 19, antiderivative size = 100 \[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\frac {x^2 \operatorname {PolyLog}\left (1+n,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 x \operatorname {PolyLog}\left (2+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {2 \operatorname {PolyLog}\left (3+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)} \]
x^2*polylog(1+n,d*(F^(c*(b*x+a)))^p)/b/c/p/ln(F)-2*x*polylog(2+n,d*(F^(c*( b*x+a)))^p)/b^2/c^2/p^2/ln(F)^2+2*polylog(3+n,d*(F^(c*(b*x+a)))^p)/b^3/c^3 /p^3/ln(F)^3
Time = 0.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\frac {x^2 \operatorname {PolyLog}\left (1+n,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 x \operatorname {PolyLog}\left (2+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {2 \operatorname {PolyLog}\left (3+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)} \]
(x^2*PolyLog[1 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - (2*x*PolyLog[ 2 + n, d*(F^(c*(a + b*x)))^p])/(b^2*c^2*p^2*Log[F]^2) + (2*PolyLog[3 + n, d*(F^(c*(a + b*x)))^p])/(b^3*c^3*p^3*Log[F]^3)
Time = 0.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {7163, 7163, 2720, 7143}
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.
\(\displaystyle \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {x^2 \operatorname {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 \int x \operatorname {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )dx}{b c p \log (F)}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {x^2 \operatorname {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\int \operatorname {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )dx}{b c p \log (F)}\right )}{b c p \log (F)}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {x^2 \operatorname {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\int F^{-c (a+b x)} \operatorname {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )dF^{c (a+b x)}}{b^2 c^2 p \log ^2(F)}\right )}{b c p \log (F)}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {x^2 \operatorname {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {2 \left (\frac {x \operatorname {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\operatorname {PolyLog}\left (n+3,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}\right )}{b c p \log (F)}\) |
(x^2*PolyLog[1 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - (2*((x*PolyLo g[2 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - PolyLog[3 + n, d*(F^(c*( a + b*x)))^p]/(b^2*c^2*p^2*Log[F]^2)))/(b*c*p*Log[F])
3.2.57.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int x^{2} \operatorname {polylog}\left (n , d \left (F^{c \left (b x +a \right )}\right )^{p}\right )d x\]
\[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\int { x^{2} {\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d) \,d x } \]
\[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\int x^{2} \operatorname {Li}_{n}\left (d \left (F^{a c + b c x}\right )^{p}\right )\, dx \]
\[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\int { x^{2} {\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d) \,d x } \]
\[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\int { x^{2} {\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d) \,d x } \]
Timed out. \[ \int x^2 \operatorname {PolyLog}\left (n,d \left (F^{c (a+b x)}\right )^p\right ) \, dx=\int x^2\,\mathrm {polylog}\left (n,d\,{\left (F^{c\,\left (a+b\,x\right )}\right )}^p\right ) \,d x \]