Integrand size = 17, antiderivative size = 217 \[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=-\frac {x^3}{a^2 d}+\frac {x^3}{a d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^2}+\frac {3 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^2 d^2}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^2 d}+\frac {6 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^2 d^3}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^2 d^2}-\frac {6 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^2 d^4}+\frac {6 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^2 d^3}-\frac {6 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a}\right )}{a^2 d^4} \]
-x^3/a^2/d+x^3/a/d/(a+b*exp(d*x+c))+1/4*x^4/a^2+3*x^2*ln(1+b*exp(d*x+c)/a) /a^2/d^2-x^3*ln(1+b*exp(d*x+c)/a)/a^2/d+6*x*polylog(2,-b*exp(d*x+c)/a)/a^2 /d^3-3*x^2*polylog(2,-b*exp(d*x+c)/a)/a^2/d^2-6*polylog(3,-b*exp(d*x+c)/a) /a^2/d^4+6*x*polylog(3,-b*exp(d*x+c)/a)/a^2/d^3-6*polylog(4,-b*exp(d*x+c)/ a)/a^2/d^4
Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\frac {-\frac {4 x^3}{d}+\frac {4 a x^3}{a d+b d e^{c+d x}}+x^4+\frac {12 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {4 x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{d}-\frac {12 x (-2+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d^3}+\frac {24 (-1+d x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^4}-\frac {24 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a}\right )}{d^4}}{4 a^2} \]
((-4*x^3)/d + (4*a*x^3)/(a*d + b*d*E^(c + d*x)) + x^4 + (12*x^2*Log[1 + (b *E^(c + d*x))/a])/d^2 - (4*x^3*Log[1 + (b*E^(c + d*x))/a])/d - (12*x*(-2 + d*x)*PolyLog[2, -((b*E^(c + d*x))/a)])/d^3 + (24*(-1 + d*x)*PolyLog[3, -( (b*E^(c + d*x))/a)])/d^4 - (24*PolyLog[4, -((b*E^(c + d*x))/a)])/d^4)/(4*a ^2)
Time = 1.66 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {2616, 2615, 2620, 2621, 2615, 2620, 3011, 2720, 7143, 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.
\(\displaystyle \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx\) |
\(\Big \downarrow \) 2616 |
\(\displaystyle \frac {\int \frac {x^3}{a+b e^{c+d x}}dx}{a}-\frac {b \int \frac {e^{c+d x} x^3}{\left (a+b e^{c+d x}\right )^2}dx}{a}\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \int \frac {e^{c+d x} x^3}{a+b e^{c+d x}}dx}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^3}{\left (a+b e^{c+d x}\right )^2}dx}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \int \frac {e^{c+d x} x^3}{\left (a+b e^{c+d x}\right )^2}dx}{a}\) |
\(\Big \downarrow \) 2621 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \int \frac {x^2}{a+b e^{c+d x}}dx}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}}dx}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a}+1\right )dx}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )dx}{d}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d}-\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {x^4}{4 a}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d}-\frac {\operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a}\right )}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}}{a}-\frac {b \left (\frac {3 \left (\frac {x^3}{3 a}-\frac {b \left (\frac {x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{d}\right )}{b d}\right )}{a}\right )}{b d}-\frac {x^3}{b d \left (a+b e^{c+d x}\right )}\right )}{a}\) |
-((b*(-(x^3/(b*d*(a + b*E^(c + d*x)))) + (3*(x^3/(3*a) - (b*((x^2*Log[1 + (b*E^(c + d*x))/a])/(b*d) - (2*(-((x*PolyLog[2, -((b*E^(c + d*x))/a)])/d) + PolyLog[3, -((b*E^(c + d*x))/a)]/d^2))/(b*d)))/a))/(b*d)))/a) + (x^4/(4* a) - (b*((x^3*Log[1 + (b*E^(c + d*x))/a])/(b*d) - (3*(-((x^2*PolyLog[2, -( (b*E^(c + d*x))/a)])/d) + (2*((x*PolyLog[3, -((b*E^(c + d*x))/a)])/d - Pol yLog[4, -((b*E^(c + d*x))/a)]/d^2))/d))/(b*d)))/a)/a
3.1.9.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[1/a Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Simp[b/a Int[(c + d*x)^m*(F^(g*(e + f*x)))^ n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n }, x] && ILtQ[p, 0] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*( (e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)*Log [F])), x] - Simp[d*(m/(b*f*g*n*(p + 1)*Log[F])) Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]
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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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]
Time = 0.09 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\frac {x^{3}}{a d \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {c^{3} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{4} a^{2}}-\frac {c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{4} a^{2}}+\frac {x^{4}}{4 a^{2}}-\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c^{3}}{d^{4} a^{2}}+\frac {2 c^{3}}{d^{4} a^{2}}-\frac {x^{3}}{a^{2} d}-\frac {3 \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c^{2}}{d^{4} a^{2}}+\frac {c^{3} x}{d^{3} a^{2}}-\frac {x^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d}-\frac {3 x^{2} \operatorname {Li}_{2}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{2}}+\frac {6 x \,\operatorname {Li}_{3}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{3}}+\frac {3 c^{2} x}{d^{3} a^{2}}+\frac {3 x^{2} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{2}}+\frac {6 x \,\operatorname {Li}_{2}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{3}}-\frac {6 \,\operatorname {Li}_{3}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{4}}+\frac {3 c^{2} \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{4} a^{2}}-\frac {3 c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{4} a^{2}}+\frac {3 c^{4}}{4 d^{4} a^{2}}-\frac {6 \,\operatorname {Li}_{4}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{2} d^{4}}\) | \(382\) |
x^3/a/d/(a+b*exp(d*x+c))+1/d^4/a^2*c^3*ln(a+b*exp(d*x+c))-1/d^4/a^2*c^3*ln (exp(d*x+c))+1/4*x^4/a^2-1/d^4/a^2*ln(1+b*exp(d*x+c)/a)*c^3+2/d^4/a^2*c^3- x^3/a^2/d-3/d^4/a^2*ln(1+b*exp(d*x+c)/a)*c^2+1/d^3/a^2*c^3*x-x^3*ln(1+b*ex p(d*x+c)/a)/a^2/d-3*x^2*polylog(2,-b*exp(d*x+c)/a)/a^2/d^2+6*x*polylog(3,- b*exp(d*x+c)/a)/a^2/d^3+3/d^3/a^2*c^2*x+3*x^2*ln(1+b*exp(d*x+c)/a)/a^2/d^2 +6*x*polylog(2,-b*exp(d*x+c)/a)/a^2/d^3-6*polylog(3,-b*exp(d*x+c)/a)/a^2/d ^4+3/d^4/a^2*c^2*ln(a+b*exp(d*x+c))-3/d^4/a^2*c^2*ln(exp(d*x+c))+3/4/d^4/a ^2*c^4-6*polylog(4,-b*exp(d*x+c)/a)/a^2/d^4
Time = 0.26 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.52 \[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\frac {a d^{4} x^{4} - a c^{4} - 4 \, a c^{3} - 12 \, {\left (a d^{2} x^{2} - 2 \, a d x + {\left (b d^{2} x^{2} - 2 \, b d x\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + {\left (b d^{4} x^{4} - 4 \, b d^{3} x^{3} - b c^{4} - 4 \, b c^{3}\right )} e^{\left (d x + c\right )} + 4 \, {\left (a c^{3} + 3 \, a c^{2} + {\left (b c^{3} + 3 \, b c^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 4 \, {\left (a d^{3} x^{3} - 3 \, a d^{2} x^{2} + a c^{3} + 3 \, a c^{2} + {\left (b d^{3} x^{3} - 3 \, b d^{2} x^{2} + b c^{3} + 3 \, b c^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) - 24 \, {\left (b e^{\left (d x + c\right )} + a\right )} {\rm polylog}\left (4, -\frac {b e^{\left (d x + c\right )}}{a}\right ) + 24 \, {\left (a d x + {\left (b d x - b\right )} e^{\left (d x + c\right )} - a\right )} {\rm polylog}\left (3, -\frac {b e^{\left (d x + c\right )}}{a}\right )}{4 \, {\left (a^{2} b d^{4} e^{\left (d x + c\right )} + a^{3} d^{4}\right )}} \]
1/4*(a*d^4*x^4 - a*c^4 - 4*a*c^3 - 12*(a*d^2*x^2 - 2*a*d*x + (b*d^2*x^2 - 2*b*d*x)*e^(d*x + c))*dilog(-(b*e^(d*x + c) + a)/a + 1) + (b*d^4*x^4 - 4*b *d^3*x^3 - b*c^4 - 4*b*c^3)*e^(d*x + c) + 4*(a*c^3 + 3*a*c^2 + (b*c^3 + 3* b*c^2)*e^(d*x + c))*log(b*e^(d*x + c) + a) - 4*(a*d^3*x^3 - 3*a*d^2*x^2 + a*c^3 + 3*a*c^2 + (b*d^3*x^3 - 3*b*d^2*x^2 + b*c^3 + 3*b*c^2)*e^(d*x + c)) *log((b*e^(d*x + c) + a)/a) - 24*(b*e^(d*x + c) + a)*polylog(4, -b*e^(d*x + c)/a) + 24*(a*d*x + (b*d*x - b)*e^(d*x + c) - a)*polylog(3, -b*e^(d*x + c)/a))/(a^2*b*d^4*e^(d*x + c) + a^3*d^4)
\[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\frac {x^{3}}{a^{2} d + a b d e^{c + d x}} + \frac {\int \left (- \frac {3 x^{2}}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac {d x^{3}}{a + b e^{c} e^{d x}}\, dx}{a d} \]
x**3/(a**2*d + a*b*d*exp(c + d*x)) + (Integral(-3*x**2/(a + b*exp(c)*exp(d *x)), x) + Integral(d*x**3/(a + b*exp(c)*exp(d*x)), x))/(a*d)
Time = 0.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\frac {x^{3}}{a b d e^{\left (d x + c\right )} + a^{2} d} + \frac {d^{4} x^{4} - 4 \, d^{3} x^{3}}{4 \, a^{2} d^{4}} - \frac {d^{3} x^{3} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 6 \, d x {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a}) + 6 \, {\rm Li}_{4}(-\frac {b e^{\left (d x + c\right )}}{a})}{a^{2} d^{4}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 2 \, {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a})\right )}}{a^{2} d^{4}} \]
x^3/(a*b*d*e^(d*x + c) + a^2*d) + 1/4*(d^4*x^4 - 4*d^3*x^3)/(a^2*d^4) - (d ^3*x^3*log(b*e^(d*x + c)/a + 1) + 3*d^2*x^2*dilog(-b*e^(d*x + c)/a) - 6*d* x*polylog(3, -b*e^(d*x + c)/a) + 6*polylog(4, -b*e^(d*x + c)/a))/(a^2*d^4) + 3*(d^2*x^2*log(b*e^(d*x + c)/a + 1) + 2*d*x*dilog(-b*e^(d*x + c)/a) - 2 *polylog(3, -b*e^(d*x + c)/a))/(a^2*d^4)
\[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b e^{\left (d x + c\right )} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx=\int \frac {x^3}{{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^2} \,d x \]