3.82 \(\int \frac{F^{c+d x} x^3}{(a+b F^{c+d x})^2} \, dx\)

Optimal. Leaf size=140 \[ -\frac{6 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}+\frac{6 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a b d^4 \log ^4(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^3}{a b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.247519, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2191, 2184, 2190, 2531, 2282, 6589} \[ -\frac{6 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}+\frac{6 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a b d^4 \log ^4(F)}-\frac{3 x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a b d^2 \log ^2(F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )}+\frac{x^3}{a b d \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{F^{c+d x} x^3}{\left (a+b F^{c+d x}\right )^2} \, dx &=-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}+\frac{3 \int \frac{x^2}{a+b F^{c+d x}} \, dx}{b d \log (F)}\\ &=\frac{x^3}{a b d \log (F)}-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac{3 \int \frac{F^{c+d x} x^2}{a+b F^{c+d x}} \, dx}{a d \log (F)}\\ &=\frac{x^3}{a b d \log (F)}-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}+\frac{6 \int x \log \left (1+\frac{b F^{c+d x}}{a}\right ) \, dx}{a b d^2 \log ^2(F)}\\ &=\frac{x^3}{a b d \log (F)}-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}-\frac{6 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}+\frac{6 \int \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right ) \, dx}{a b d^3 \log ^3(F)}\\ &=\frac{x^3}{a b d \log (F)}-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}-\frac{6 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,x,F^{c+d x}\right )}{a b d^4 \log ^4(F)}\\ &=\frac{x^3}{a b d \log (F)}-\frac{x^3}{b d \left (a+b F^{c+d x}\right ) \log (F)}-\frac{3 x^2 \log \left (1+\frac{b F^{c+d x}}{a}\right )}{a b d^2 \log ^2(F)}-\frac{6 x \text{Li}_2\left (-\frac{b F^{c+d x}}{a}\right )}{a b d^3 \log ^3(F)}+\frac{6 \text{Li}_3\left (-\frac{b F^{c+d x}}{a}\right )}{a b d^4 \log ^4(F)}\\ \end{align*}

Mathematica [A]  time = 0.157075, size = 137, normalized size = 0.98 \[ \frac{3 \left (-\frac{2 x \text{PolyLog}\left (2,-\frac{b F^{c+d x}}{a}\right )}{a d^2 \log ^2(F)}+\frac{2 \text{PolyLog}\left (3,-\frac{b F^{c+d x}}{a}\right )}{a d^3 \log ^3(F)}-\frac{x^2 \log \left (\frac{b F^{c+d x}}{a}+1\right )}{a d \log (F)}+\frac{x^3}{3 a}\right )}{b d \log (F)}-\frac{x^3}{b d \log (F) \left (a+b F^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(F^(c + d*x)*x^3)/(a + b*F^(c + d*x))^2,x]

[Out]

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

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Maple [A]  time = 0.03, size = 274, normalized size = 2. \begin{align*} -{\frac{{x}^{3}}{bd \left ( a+b{F}^{dx+c} \right ) \ln \left ( F \right ) }}+{\frac{{x}^{3}}{\ln \left ( F \right ) abd}}-3\,{\frac{{c}^{2}x}{b{d}^{3}\ln \left ( F \right ) a}}-2\,{\frac{{c}^{3}}{b{d}^{4}\ln \left ( F \right ) a}}-3\,{\frac{{x}^{2}}{b{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}a}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+3\,{\frac{{c}^{2}}{b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}a}\ln \left ( 1+{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }-6\,{\frac{x}{b{d}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}a}{\it polylog} \left ( 2,-{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }+6\,{\frac{1}{b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}a}{\it polylog} \left ( 3,-{\frac{b{F}^{dx}{F}^{c}}{a}} \right ) }-3\,{\frac{{c}^{2}\ln \left ( a+b{F}^{dx}{F}^{c} \right ) }{b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}a}}+3\,{\frac{{c}^{2}\ln \left ({F}^{dx}{F}^{c} \right ) }{b{d}^{4} \left ( \ln \left ( F \right ) \right ) ^{2}a}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.15233, size = 181, normalized size = 1.29 \begin{align*} -\frac{x^{3}}{F^{d x} F^{c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} + \frac{\log \left (F^{d x}\right )^{3}}{a 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 )^{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 )}}{a 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 1.55304, size = 575, normalized size = 4.11 \begin{align*} \frac{a c^{3} \log \left (F\right )^{3} +{\left (b d^{3} x^{3} + b c^{3}\right )} F^{d x + c} \log \left (F\right )^{3} - 6 \,{\left (F^{d x + c} b d x \log \left (F\right ) + a d x \log \left (F\right )\right )}{\rm Li}_2\left (-\frac{F^{d x + c} b + a}{a} + 1\right ) - 3 \,{\left (F^{d x + c} b c^{2} \log \left (F\right )^{2} + a c^{2} \log \left (F\right )^{2}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \,{\left ({\left (b d^{2} x^{2} - b c^{2}\right )} F^{d x + c} \log \left (F\right )^{2} +{\left (a d^{2} x^{2} - a c^{2}\right )} \log \left (F\right )^{2}\right )} \log \left (\frac{F^{d x + c} b + a}{a}\right ) + 6 \,{\left (F^{d x + c} b + a\right )}{\rm polylog}\left (3, -\frac{F^{d x + c} b}{a}\right )}{F^{d x + c} a b^{2} d^{4} \log \left (F\right )^{4} + 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))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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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 )}^{2}}\,{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))^2,x, algorithm="giac")

[Out]

integrate(F^(d*x + c)*x^3/(F^(d*x + c)*b + a)^2, x)