3.88 \(\int \frac{F^{c+d x} x^3}{(a+b F^{c+d x})^3} \, dx\)

Optimal. Leaf size=261 \[ -\frac{3 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{3 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac{3 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}+\frac{3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^3}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

(-3*x^2)/(2*a^2*b*d^2*Log[F]^2) + (3*x^2)/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x^3/(2*a^2*b*d*Log[F]) -
x^3/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) + (3*x*Log[1 + (b*F^(c + d*x))/a])/(a^2*b*d^3*Log[F]^3) - (3*x^2*Log[
1 + (b*F^(c + d*x))/a])/(2*a^2*b*d^2*Log[F]^2) + (3*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*Log[F]^4) - (
3*x*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^3*Log[F]^3) + (3*PolyLog[3, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*L
og[F]^4)

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Rubi [A]  time = 0.503632, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2191, 2185, 2184, 2190, 2531, 2282, 6589, 2279, 2391} \[ -\frac{3 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{3 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac{3 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}+\frac{3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x^3}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^3,x]

[Out]

(-3*x^2)/(2*a^2*b*d^2*Log[F]^2) + (3*x^2)/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x^3/(2*a^2*b*d*Log[F]) -
x^3/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) + (3*x*Log[1 + (b*F^(c + d*x))/a])/(a^2*b*d^3*Log[F]^3) - (3*x^2*Log[
1 + (b*F^(c + d*x))/a])/(2*a^2*b*d^2*Log[F]^2) + (3*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*Log[F]^4) - (
3*x*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^3*Log[F]^3) + (3*PolyLog[3, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*L
og[F]^4)

Rule 2191

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] - Dist[(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]

Rule 2185

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[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
]

Rule 2184

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] - Dist[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]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

Rule 2531

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] + Dist[(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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{F^{c+d x} x^3}{\left (a+b F^{c+d x}\right )^3} \, dx &=-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac{3 \int \frac{x^2}{\left (a+b F^{c+d x}\right )^2} \, dx}{2 b d \log (F)}\\ &=-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac{3 \int \frac{F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^2} \, dx}{2 a d \log (F)}+\frac{3 \int \frac{x^2}{a+b F^{c+d x}} \, dx}{2 a b d \log (F)}\\ &=\frac{3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac{3 \int \frac{x}{a+b F^{c+d x}} \, dx}{a b d^2 \log ^2(F)}-\frac{3 \int \frac{F^{c+d x} x^2}{a+b F^{c+d x}} \, dx}{2 a^2 d \log (F)}\\ &=-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{3 \int \frac{F^{c+d x} x}{a+b F^{c+d x}} \, dx}{a^2 d^2 \log ^2(F)}+\frac{3 \int x \log \left (1+\frac{b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^2 \log ^2(F)}\\ &=-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac{3 x \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}-\frac{3 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 \int \log \left (1+\frac{b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^3 \log ^3(F)}+\frac{3 \int \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right ) \, dx}{a^2 b d^3 \log ^3(F)}\\ &=-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac{3 x \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}-\frac{3 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^4 \log ^4(F)}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a^2 b d^4 \log ^4(F)}\\ &=-\frac{3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac{3 x^2}{2 a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac{x^3}{2 a^2 b d \log (F)}-\frac{x^3}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac{3 x \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{3 \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac{3 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac{3 \text{Li}_3\left (-\frac{b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}\\ \end{align*}

Mathematica [A]  time = 0.323482, size = 220, normalized size = 0.84 \[ \frac{6 \left (a+b F^{c+d x}\right )^2 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )-6 (d x \log (F)-1) \left (a+b F^{c+d x}\right )^2 \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )+d x \log (F) \left (b d^2 x^2 \log ^2(F) F^{c+d x} \left (2 a+b F^{c+d x}\right )+6 \left (a+b F^{c+d x}\right )^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )-3 d x \log (F) \left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (\frac{b F^{c+d x}}{a}+1\right )+b F^{c+d x}\right )\right )}{2 a^2 b d^4 \log ^4(F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^3,x]

[Out]

(d*x*Log[F]*(b*d^2*F^(c + d*x)*(2*a + b*F^(c + d*x))*x^2*Log[F]^2 + 6*(a + b*F^(c + d*x))^2*Log[1 + (b*F^(c +
d*x))/a] - 3*d*(a + b*F^(c + d*x))*x*Log[F]*(b*F^(c + d*x) + (a + b*F^(c + d*x))*Log[1 + (b*F^(c + d*x))/a]))
- 6*(a + b*F^(c + d*x))^2*(-1 + d*x*Log[F])*PolyLog[2, -((b*F^(c + d*x))/a)] + 6*(a + b*F^(c + d*x))^2*PolyLog
[3, -((b*F^(c + d*x))/a)])/(2*a^2*b*d^4*(a + b*F^(c + d*x))^2*Log[F]^4)

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Maple [A]  time = 0.066, size = 501, normalized size = 1.9 \begin{align*} -{\frac{{x}^{2} \left ( \ln \left ( F \right ) adx-3\,b{F}^{dx+c}-3\,a \right ) }{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}b \left ( a+b{F}^{dx+c} \right ) ^{2}a}}+{\frac{{x}^{3}}{2\,{a}^{2}bd\ln \left ( F \right ) }}-{\frac{3\,{c}^{2}x}{2\,\ln \left ( F \right ){d}^{3}{a}^{2}b}}-{\frac{{c}^{3}}{\ln \left ( F \right ){d}^{4}{a}^{2}b}}-{\frac{3\,{x}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}{a}^{2}b}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+{\frac{3\,{c}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}{a}^{2}b}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }-3\,{\frac{x}{ \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{3}{a}^{2}b}{\it polylog} \left ( 2,-{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+3\,{\frac{1}{ \left ( \ln \left ( F \right ) \right ) ^{4}{d}^{4}{a}^{2}b}{\it polylog} \left ( 3,-{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }-{\frac{3\,{c}^{2}\ln \left ( a+b{F}^{dx}{F}^{c} \right ) }{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}{a}^{2}b}}+{\frac{3\,{c}^{2}\ln \left ({F}^{dx}{F}^{c} \right ) }{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}{a}^{2}b}}-{\frac{3\,{x}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}{a}^{2}b}}-3\,{\frac{cx}{ \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{3}{a}^{2}b}}-{\frac{3\,{c}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{4}{a}^{2}b}}+3\,{\frac{x}{ \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{3}{a}^{2}b}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+3\,{\frac{c}{ \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{4}{a}^{2}b}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+3\,{\frac{1}{ \left ( \ln \left ( F \right ) \right ) ^{4}{d}^{4}{a}^{2}b}{\it polylog} \left ( 2,-{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }-3\,{\frac{c\ln \left ( a+b{F}^{dx}{F}^{c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{4}{a}^{2}b}}+3\,{\frac{c\ln \left ({F}^{dx}{F}^{c} \right ) }{ \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{4}{a}^{2}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(d*x+c)*x^3/(a+b*F^(d*x+c))^3,x)

[Out]

-1/2*x^2*(ln(F)*a*d*x-3*b*F^(d*x+c)-3*a)/ln(F)^2/d^2/b/(a+b*F^(d*x+c))^2/a+1/2*x^3/a^2/b/d/ln(F)-3/2/ln(F)/d^3
/a^2/b*c^2*x-1/ln(F)/d^4/a^2/b*c^3-3/2/ln(F)^2/d^2/a^2/b*ln(1+b*F^(d*x)*F^c/a)*x^2+3/2/ln(F)^2/d^4/a^2/b*ln(1+
b*F^(d*x)*F^c/a)*c^2-3/ln(F)^3/d^3/a^2/b*polylog(2,-b*F^(d*x)*F^c/a)*x+3/ln(F)^4/d^4/a^2/b*polylog(3,-b*F^(d*x
)*F^c/a)-3/2/ln(F)^2/d^4/a^2/b*c^2*ln(a+b*F^(d*x)*F^c)+3/2/ln(F)^2/d^4/a^2/b*c^2*ln(F^(d*x)*F^c)-3/2*x^2/a^2/b
/d^2/ln(F)^2-3/ln(F)^2/d^3/a^2/b*c*x-3/2/ln(F)^2/d^4/a^2/b*c^2+3/ln(F)^3/d^3/a^2/b*ln(1+b*F^(d*x)*F^c/a)*x+3/l
n(F)^3/d^4/a^2/b*ln(1+b*F^(d*x)*F^c/a)*c+3/ln(F)^4/d^4/a^2/b*polylog(2,-b*F^(d*x)*F^c/a)-3/ln(F)^3/d^4/a^2/b*c
*ln(a+b*F^(d*x)*F^c)+3/ln(F)^3/d^4/a^2/b*c*ln(F^(d*x)*F^c)

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Maxima [A]  time = 1.14113, size = 355, normalized size = 1.36 \begin{align*} -\frac{a d x^{3} \log \left (F\right ) - 3 \, F^{d x} F^{c} b x^{2} - 3 \, a x^{2}}{2 \,{\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \left (F\right )^{2} + a^{3} b d^{2} \log \left (F\right )^{2}\right )}} - \frac{3 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{2} + 2 \,{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right ) - 2 \,{\rm Li}_{3}(-\frac{F^{d x} F^{c} b}{a})\right )}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} + \frac{\log \left (F^{d x}\right )^{3} - 3 \, \log \left (F^{d x}\right )^{2}}{2 \, a^{2} b d^{4} \log \left (F\right )^{4}} + \frac{3 \,{\left (\log \left (\frac{F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) +{\rm Li}_2\left (-\frac{F^{d x} F^{c} b}{a}\right )\right )}}{a^{2} b d^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)*x^3/(a+b*F^(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*(a*d*x^3*log(F) - 3*F^(d*x)*F^c*b*x^2 - 3*a*x^2)/(2*F^(d*x)*F^c*a^2*b^2*d^2*log(F)^2 + F^(2*d*x)*F^(2*c)*
a*b^3*d^2*log(F)^2 + a^3*b*d^2*log(F)^2) - 3/2*(log(F^(d*x)*F^c*b/a + 1)*log(F^(d*x))^2 + 2*dilog(-F^(d*x)*F^c
*b/a)*log(F^(d*x)) - 2*polylog(3, -F^(d*x)*F^c*b/a))/(a^2*b*d^4*log(F)^4) + 1/2*(log(F^(d*x))^3 - 3*log(F^(d*x
))^2)/(a^2*b*d^4*log(F)^4) + 3*(log(F^(d*x)*F^c*b/a + 1)*log(F^(d*x)) + dilog(-F^(d*x)*F^c*b/a))/(a^2*b*d^4*lo
g(F)^4)

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Fricas [C]  time = 1.63696, size = 1310, normalized size = 5.02 \begin{align*} \frac{a^{2} c^{3} \log \left (F\right )^{3} + 3 \, a^{2} c^{2} \log \left (F\right )^{2} +{\left ({\left (b^{2} d^{3} x^{3} + b^{2} c^{3}\right )} \log \left (F\right )^{3} - 3 \,{\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2}\right )} F^{2 \, d x + 2 \, c} +{\left (2 \,{\left (a b d^{3} x^{3} + a b c^{3}\right )} \log \left (F\right )^{3} - 3 \,{\left (a b d^{2} x^{2} - 2 \, a b c^{2}\right )} \log \left (F\right )^{2}\right )} F^{d x + c} - 6 \,{\left (a^{2} d x \log \left (F\right ) +{\left (b^{2} d x \log \left (F\right ) - b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left (a b d x \log \left (F\right ) - a b\right )} F^{d x + c} - a^{2}\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 3 \,{\left (a^{2} c^{2} \log \left (F\right )^{2} + 2 \, a^{2} c \log \left (F\right ) +{\left (b^{2} c^{2} \log \left (F\right )^{2} + 2 \, b^{2} c \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left (a b c^{2} \log \left (F\right )^{2} + 2 \, a b c \log \left (F\right )\right )} F^{d x + c}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \,{\left ({\left (a^{2} d^{2} x^{2} - a^{2} c^{2}\right )} \log \left (F\right )^{2} +{\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} + 2 \,{\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (a b d x + a b c\right )} \log \left (F\right )\right )} F^{d x + c} - 2 \,{\left (a^{2} d x + a^{2} c\right )} \log \left (F\right )\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right ) + 6 \,{\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )}{\rm polylog}\left (3, -\frac{F^{d x + c} b}{a}\right )}{2 \,{\left (2 \, F^{d x + c} a^{3} b^{2} d^{4} \log \left (F\right )^{4} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{4} \log \left (F\right )^{4} + a^{4} b d^{4} \log \left (F\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)*x^3/(a+b*F^(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2*(a^2*c^3*log(F)^3 + 3*a^2*c^2*log(F)^2 + ((b^2*d^3*x^3 + b^2*c^3)*log(F)^3 - 3*(b^2*d^2*x^2 - b^2*c^2)*log
(F)^2)*F^(2*d*x + 2*c) + (2*(a*b*d^3*x^3 + a*b*c^3)*log(F)^3 - 3*(a*b*d^2*x^2 - 2*a*b*c^2)*log(F)^2)*F^(d*x +
c) - 6*(a^2*d*x*log(F) + (b^2*d*x*log(F) - b^2)*F^(2*d*x + 2*c) + 2*(a*b*d*x*log(F) - a*b)*F^(d*x + c) - a^2)*
dilog(-(F^(d*x + c)*b + a)/a + 1) - 3*(a^2*c^2*log(F)^2 + 2*a^2*c*log(F) + (b^2*c^2*log(F)^2 + 2*b^2*c*log(F))
*F^(2*d*x + 2*c) + 2*(a*b*c^2*log(F)^2 + 2*a*b*c*log(F))*F^(d*x + c))*log(F^(d*x + c)*b + a) - 3*((a^2*d^2*x^2
- a^2*c^2)*log(F)^2 + ((b^2*d^2*x^2 - b^2*c^2)*log(F)^2 - 2*(b^2*d*x + b^2*c)*log(F))*F^(2*d*x + 2*c) + 2*((a
*b*d^2*x^2 - a*b*c^2)*log(F)^2 - 2*(a*b*d*x + a*b*c)*log(F))*F^(d*x + c) - 2*(a^2*d*x + a^2*c)*log(F))*log((F^
(d*x + c)*b + a)/a) + 6*(2*F^(d*x + c)*a*b + F^(2*d*x + 2*c)*b^2 + a^2)*polylog(3, -F^(d*x + c)*b/a))/(2*F^(d*
x + c)*a^3*b^2*d^4*log(F)^4 + F^(2*d*x + 2*c)*a^2*b^3*d^4*log(F)^4 + a^4*b*d^4*log(F)^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 F^{c + d x} b x^{2} - a d x^{3} \log{\left (F \right )} + 3 a x^{2}}{4 F^{c + d x} a^{2} b^{2} d^{2} \log{\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log{\left (F \right )}^{2} + 2 a^{3} b d^{2} \log{\left (F \right )}^{2}} + \frac{3 \left (\int - \frac{2 x}{a + b e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx + \int \frac{d x^{2} \log{\left (F \right )}}{a + b e^{c \log{\left (F \right )}} e^{d x \log{\left (F \right )}}}\, dx\right )}{2 a b d^{2} \log{\left (F \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(d*x+c)*x**3/(a+b*F**(d*x+c))**3,x)

[Out]

(3*F**(c + d*x)*b*x**2 - a*d*x**3*log(F) + 3*a*x**2)/(4*F**(c + d*x)*a**2*b**2*d**2*log(F)**2 + 2*F**(2*c + 2*
d*x)*a*b**3*d**2*log(F)**2 + 2*a**3*b*d**2*log(F)**2) + 3*(Integral(-2*x/(a + b*exp(c*log(F))*exp(d*x*log(F)))
, x) + Integral(d*x**2*log(F)/(a + b*exp(c*log(F))*exp(d*x*log(F))), x))/(2*a*b*d**2*log(F)**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{d x + c} x^{3}}{{\left (F^{d x + c} b + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)*x^3/(a+b*F^(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^3, x)