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

Optimal. Leaf size=261 \[ \frac {3 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}+\frac {3 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^4 \log ^4(F)}-\frac {3 x \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)}+\frac {3 x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{a^2 b d^3 \log ^3(F)}-\frac {3 x^2 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{2 a^2 b d^2 \log ^2(F)}-\frac {3 x^2}{2 a^2 b d^2 \log ^2(F)}+\frac {x^3}{2 a^2 b d \log (F)}+\frac {3 x^2}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac {x^3}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 261, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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 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]

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 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 )^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*}

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Mathematica [A]  time = 0.32, size = 220, normalized size = 0.84 \[ \frac {d x \log (F) \left (b d^2 x^2 \log ^2(F) F^{c+d x} \left (2 a+b F^{c+d x}\right )+6 \left (a+b F^{c+d x}\right )^2 \log \left (\frac {b F^{c+d x}}{a}+1\right )-3 d x \log (F) \left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (\frac {b F^{c+d x}}{a}+1\right )+b F^{c+d x}\right )\right )+6 \left (a+b F^{c+d x}\right )^2 \text {Li}_3\left (-\frac {b F^{c+d x}}{a}\right )-6 (d x \log (F)-1) \left (a+b F^{c+d x}\right )^2 \text {Li}_2\left (-\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^4 \log ^4(F) \left (a+b F^{c+d x}\right )^2} \]

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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fricas [C]  time = 0.46, size = 577, normalized size = 2.21 \[ \frac {a^{2} c^{3} \log \relax (F)^{3} + 3 \, a^{2} c^{2} \log \relax (F)^{2} + {\left ({\left (b^{2} d^{3} x^{3} + b^{2} c^{3}\right )} \log \relax (F)^{3} - 3 \, {\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \relax (F)^{2}\right )} F^{2 \, d x + 2 \, c} + {\left (2 \, {\left (a b d^{3} x^{3} + a b c^{3}\right )} \log \relax (F)^{3} - 3 \, {\left (a b d^{2} x^{2} - 2 \, a b c^{2}\right )} \log \relax (F)^{2}\right )} F^{d x + c} - 6 \, {\left (a^{2} d x \log \relax (F) + {\left (b^{2} d x \log \relax (F) - b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b d x \log \relax (F) - a b\right )} F^{d x + c} - a^{2}\right )} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) - 3 \, {\left (a^{2} c^{2} \log \relax (F)^{2} + 2 \, a^{2} c \log \relax (F) + {\left (b^{2} c^{2} \log \relax (F)^{2} + 2 \, b^{2} c \log \relax (F)\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b c^{2} \log \relax (F)^{2} + 2 \, a b c \log \relax (F)\right )} F^{d x + c}\right )} \log \left (F^{d x + c} b + a\right ) - 3 \, {\left ({\left (a^{2} d^{2} x^{2} - a^{2} c^{2}\right )} \log \relax (F)^{2} + {\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (b^{2} d x + b^{2} c\right )} \log \relax (F)\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \relax (F)^{2} - 2 \, {\left (a b d x + a b c\right )} \log \relax (F)\right )} F^{d x + c} - 2 \, {\left (a^{2} d x + a^{2} c\right )} \log \relax (F)\right )} \log \left (\frac {F^{d x + c} b + a}{a}\right ) + 6 \, {\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} {\rm polylog}\left (3, -\frac {F^{d x + c} b}{a}\right )}{2 \, {\left (2 \, F^{d x + c} a^{3} b^{2} d^{4} \log \relax (F)^{4} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{4} \log \relax (F)^{4} + a^{4} b d^{4} \log \relax (F)^{4}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x + c} x^{3}}{{\left (F^{d x + c} b + a\right )}^{3}}\,{d x} \]

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)

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maple [A]  time = 0.08, size = 501, normalized size = 1.92 \[ \frac {x^{3}}{2 a^{2} b d \ln \relax (F )}-\frac {\left (a d x \ln \relax (F )-3 b \,F^{d x +c}-3 a \right ) x^{2}}{2 \left (b \,F^{d x +c}+a \right )^{2} a b \,d^{2} \ln \relax (F )^{2}}-\frac {3 c^{2} x}{2 a^{2} b \,d^{3} \ln \relax (F )}-\frac {3 x^{2} \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{2 a^{2} b \,d^{2} \ln \relax (F )^{2}}-\frac {c^{3}}{a^{2} b \,d^{4} \ln \relax (F )}-\frac {3 x^{2}}{2 a^{2} b \,d^{2} \ln \relax (F )^{2}}+\frac {3 c^{2} \ln \left (F^{c} F^{d x}\right )}{2 a^{2} b \,d^{4} \ln \relax (F )^{2}}+\frac {3 c^{2} \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{2 a^{2} b \,d^{4} \ln \relax (F )^{2}}-\frac {3 c^{2} \ln \left (b \,F^{c} F^{d x}+a \right )}{2 a^{2} b \,d^{4} \ln \relax (F )^{2}}-\frac {3 c x}{a^{2} b \,d^{3} \ln \relax (F )^{2}}-\frac {3 c^{2}}{2 a^{2} b \,d^{4} \ln \relax (F )^{2}}-\frac {3 x \polylog \left (2, -\frac {b \,F^{c} F^{d x}}{a}\right )}{a^{2} b \,d^{3} \ln \relax (F )^{3}}+\frac {3 x \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{a^{2} b \,d^{3} \ln \relax (F )^{3}}+\frac {3 c \ln \left (F^{c} F^{d x}\right )}{a^{2} b \,d^{4} \ln \relax (F )^{3}}+\frac {3 c \ln \left (\frac {b \,F^{c} F^{d x}}{a}+1\right )}{a^{2} b \,d^{4} \ln \relax (F )^{3}}-\frac {3 c \ln \left (b \,F^{c} F^{d x}+a \right )}{a^{2} b \,d^{4} \ln \relax (F )^{3}}+\frac {3 \polylog \left (2, -\frac {b \,F^{c} F^{d x}}{a}\right )}{a^{2} b \,d^{4} \ln \relax (F )^{4}}+\frac {3 \polylog \left (3, -\frac {b \,F^{c} F^{d x}}{a}\right )}{a^{2} b \,d^{4} \ln \relax (F )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.53, size = 263, normalized size = 1.01 \[ -\frac {a d x^{3} \log \relax (F) - 3 \, F^{d x} F^{c} b x^{2} - 3 \, a x^{2}}{2 \, {\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \relax (F)^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \relax (F)^{2} + a^{3} b d^{2} \log \relax (F)^{2}\right )}} - \frac {3 \, {\left (\log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right )^{2} + 2 \, {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right ) \log \left (F^{d x}\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{d x} F^{c} b}{a})\right )}}{2 \, a^{2} b d^{4} \log \relax (F)^{4}} + \frac {\log \left (F^{d x}\right )^{3} - 3 \, \log \left (F^{d x}\right )^{2}}{2 \, a^{2} b d^{4} \log \relax (F)^{4}} + \frac {3 \, {\left (\log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F^{d x}\right ) + {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right )\right )}}{a^{2} b d^{4} \log \relax (F)^{4}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {F^{c+d\,x}\,x^3}{{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 F^{c + d x} b x^{2} - a d x^{3} \log {\relax (F )} + 3 a x^{2}}{4 F^{c + d x} a^{2} b^{2} d^{2} \log {\relax (F )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log {\relax (F )}^{2} + 2 a^{3} b d^{2} \log {\relax (F )}^{2}} + \frac {3 \left (\int \left (- \frac {2 x}{a + b e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\right )\, dx + \int \frac {d x^{2} \log {\relax (F )}}{a + b e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx\right )}{2 a b d^{2} \log {\relax (F )}^{2}} \]

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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