3.1.46 \(\int \frac {(c+d x)^3}{a+b (F^{g (e+f x)})^n} \, dx\) [46]

Optimal. Leaf size=192 \[ \frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \text {Li}_4\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)} \]

[Out]

1/4*(d*x+c)^4/a/d-(d*x+c)^3*ln(1+b*(F^(g*(f*x+e)))^n/a)/a/f/g/n/ln(F)-3*d*(d*x+c)^2*polylog(2,-b*(F^(g*(f*x+e)
))^n/a)/a/f^2/g^2/n^2/ln(F)^2+6*d^2*(d*x+c)*polylog(3,-b*(F^(g*(f*x+e)))^n/a)/a/f^3/g^3/n^3/ln(F)^3-6*d^3*poly
log(4,-b*(F^(g*(f*x+e)))^n/a)/a/f^4/g^4/n^4/ln(F)^4

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Rubi [A]
time = 0.22, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2215, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac {6 d^3 \text {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}-\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac {(c+d x)^4}{4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^4/(4*a*d) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*Log[F]) - (3*d*(c + d*x)^2*Pol
yLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*n^2*Log[F]^2) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e +
f*x)))^n)/a)])/(a*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^4*g^4*n^4*Log[
F]^4)

Rule 2215

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 2221

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]
- 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 2320

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 2611

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 6724

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 6744

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] - Dist[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]

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx &=\frac {(c+d x)^4}{4 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a}\\ &=\frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {(3 d) \int (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f g n \log (F)}\\ &=\frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f^3 g^3 n^3 \log ^3(F)}\\ &=\frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x^n}{a}\right )}{x} \, dx,x,F^{g (e+f x)}\right )}{a f^4 g^4 n^3 \log ^4(F)}\\ &=\frac {(c+d x)^4}{4 a d}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \text {Li}_4\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^4 g^4 n^4 \log ^4(F)}\\ \end {align*}

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Mathematica [A]
time = 0.67, size = 166, normalized size = 0.86 \begin {gather*} \frac {-(c+d x)^3 \log \left (1+\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+\frac {3 d \left (f^2 g^2 n^2 (c+d x)^2 \log ^2(F) \text {Li}_2\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+2 d \left (f g n (c+d x) \log (F) \text {Li}_3\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+d \text {Li}_4\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )\right )\right )}{f^3 g^3 n^3 \log ^3(F)}}{a f g n \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3074\) vs. \(2(190)=380\).
time = 0.12, size = 3075, normalized size = 16.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(3075\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+b*(F^(g*(f*x+e)))^n),x,method=_RETURNVERBOSE)

[Out]

-1/n/g/f^4/ln(F)*d^3*e^3/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+1/n/g/f^4/ln(F)*d^3*e^3/a*ln(a+b*F^(
n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-3/n^2/g^2/f^2/ln(F)^2*d^3/a*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^
(g*(f*x+e)))^n/a)*x^2+6/n^3/g^3/f^3/ln(F)^3*d^3/a*polylog(3,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*x
-6/n/g^2/f^3/ln(F)^2*c*d^2*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+
e)))^n)+6/n/g^2/f^3/ln(F)^2*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*
(f*x+e)))^n/a)*e+6/n/g^2/f^3/ln(F)^2*c*d^2*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)
*(F^(g*(f*x+e)))^n)+3/n/g/f^2/ln(F)*c^2*d*e/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-6/g/f/ln(F)*c
*d^2/a*ln(F^(g*(f*x+e)))*x^2+6/g^2/f^2/ln(F)^2*c*d^2/a*ln(F^(g*(f*x+e)))^2*x-1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*
(f*x+e)))-g*(f*x+e)*ln(F))^3/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)-3/n^2/g^2/f^2/ln(F)^2*c^2*
d/a*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)+6/n^3/g^3/f^3/ln(F)^3*c*d^2/a*polylog(3,-b*F^(n
*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)-1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^3/a*ln(F
^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-1/n/g/f/ln(F)*d^3/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))
^n/a)*x^3-1/n/g/f^4/ln(F)*d^3/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*e^3-3/g/f/ln(F)*d^3/a*ln(
F^(g*(f*x+e)))*x^3+9/2/g^2/f^2/ln(F)^2*d^3/a*ln(F^(g*(f*x+e)))^2*x^2-3/g^3/f^3/ln(F)^3*d^3/a*ln(F^(g*(f*x+e)))
^3*x+1/g^3/f^3/ln(F)^3*d^3*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^3/a*x+3/2/g^2/f^2/ln(F)^2*c^2*d/a*ln(F^(g*(f*x+
e)))^2-3/g/f/ln(F)*c^2*d/a*ln(F^(g*(f*x+e)))*x-1/n/g/f/ln(F)*c^3/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+
e)))^n)-6/n^4/g^4/f^4/ln(F)^4*d^3/a*polylog(4,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)+1/n/g/f/ln(F)*c
^3/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+d^3/a*x^4-2/g^3/f^3/ln(F)^3*c*d^2/a*ln(F^(g*(f*x+e)))^3+3/
4/g^4/f^4/ln(F)^4*d^3/a*ln(F^(g*(f*x+e)))^4+1/f^3*d^3*e^3/a*x+3*c*d^2/a*x^3+3*c^2*d/a*x^2-3/f^2*c*d^2*e^2/a*x+
3/f*c^2*d/a*x*e+3/n/g/f^3/ln(F)*c*d^2*e^2/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-3/n/g/f^3/ln(F)*c*d
^2*e^2/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-3/n/g^2/f^2/ln(F)^2*c^2*d*(ln(F^(g*(f*x+e)))-g*(f*
x+e)*ln(F))/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+3/g/f/ln(F)*c^2*d/a*x*(ln(F^(g*(f*x+e)))-g*(f*x+e
)*ln(F))+3/g/f^3/ln(F)*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*x+3/g^2/f^3/ln(F)^2*d^3*e*(ln(F^(g*(f*x+e
)))-g*(f*x+e)*ln(F))^2/a*x-3/g^2/f^2/ln(F)^2*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*x-3/n/g^3/f^4/ln(F)
^3*d^3*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+3/n/g^3/f^4/ln
(F)^3*d^3*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-3/n/g/f
^2/ln(F)*c^2*d*e/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)-6/g/f^2/ln(F)*c*d^2*e/a*x*(ln(F^(g*(f*x+e)))
-g*(f*x+e)*ln(F))+1/n/g^4/f^4/ln(F)^4*d^3*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^3/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f
*x)*(F^(g*(f*x+e)))^n)-6/n^2/g^2/f^2/ln(F)^2*c*d^2/a*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a
)*x+3/n/g^2/f^4/ln(F)^2*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f
*x+e)))^n)+3/n/g^3/f^3/ln(F)^3*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g
*(f*x+e)))^n)-3/n/g^3/f^3/ln(F)^3*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(a+b*F^(n*g*f*x)*F^(-n*g*f*x
)*(F^(g*(f*x+e)))^n)-3/n/g^2/f^4/ln(F)^2*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(1+b*F^(n*g*f*x)*F^(-
n*g*f*x)*(F^(g*(f*x+e)))^n/a)-3/n/g^3/f^4/ln(F)^3*d^3*e*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(1+b*F^(n*g*
f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)+3/n/g^3/f^3/ln(F)^3*c*d^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))^2/a*ln(1+
b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)+3/n/g/f^3/ln(F)*c*d^2*e^2/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F
^(g*(f*x+e)))^n/a)-3/n/g^2/f^4/ln(F)^2*d^3*e^2*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(F^(n*g*f*x)*F^(-n*g*f*
x)*(F^(g*(f*x+e)))^n)+3/n/g^2/f^2/ln(F)^2*c^2*d*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))/a*ln(a+b*F^(n*g*f*x)*F^(-n
*g*f*x)*(F^(g*(f*x+e)))^n)-3/n/g/f/ln(F)*c^2*d/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*x-3/n/g/
f^2/ln(F)*c^2*d/a*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*e-3/n/g^2/f^2/ln(F)^2*c^2*d/a*ln(1+b*F^
(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*(ln(F^(g*(f*x+e)))-g*(f*x+e)*ln(F))-3/n/g/f/ln(F)*c*d^2/a*ln(1+b*F
^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*x^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (193) = 386\).
time = 0.40, size = 491, normalized size = 2.56 \begin {gather*} c^{3} {\left (\frac {f g n x + g n e}{a f g n} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a f g n \log \left (F\right )}\right )} - \frac {3 \, {\left (f g n x \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right )\right )} c^{2} d}{a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {3 \, {\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} c d^{2}}{a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (f^{3} g^{3} n^{3} x^{3} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{3} + 3 \, f^{2} g^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right )^{2} - 6 \, f g n x \log \left (F\right ) {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} d^{3}}{a f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {d^{3} f^{4} g^{4} n^{4} x^{4} \log \left (F\right )^{4} + 4 \, c d^{2} f^{4} g^{4} n^{4} x^{3} \log \left (F\right )^{4} + 6 \, c^{2} d f^{4} g^{4} n^{4} x^{2} \log \left (F\right )^{4}}{4 \, a f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*((f*g*n*x + g*n*e)/(a*f*g*n) - log(F^(f*g*n*x + g*n*e)*b + a)/(a*f*g*n*log(F))) - 3*(f*g*n*x*log(F^(f*g*n*
x)*F^(g*n*e)*b/a + 1)*log(F) + dilog(-F^(f*g*n*x)*F^(g*n*e)*b/a))*c^2*d/(a*f^2*g^2*n^2*log(F)^2) - 3*(f^2*g^2*
n^2*x^2*log(F^(f*g*n*x)*F^(g*n*e)*b/a + 1)*log(F)^2 + 2*f*g*n*x*dilog(-F^(f*g*n*x)*F^(g*n*e)*b/a)*log(F) - 2*p
olylog(3, -F^(f*g*n*x)*F^(g*n*e)*b/a))*c*d^2/(a*f^3*g^3*n^3*log(F)^3) - (f^3*g^3*n^3*x^3*log(F^(f*g*n*x)*F^(g*
n*e)*b/a + 1)*log(F)^3 + 3*f^2*g^2*n^2*x^2*dilog(-F^(f*g*n*x)*F^(g*n*e)*b/a)*log(F)^2 - 6*f*g*n*x*log(F)*polyl
og(3, -F^(f*g*n*x)*F^(g*n*e)*b/a) + 6*polylog(4, -F^(f*g*n*x)*F^(g*n*e)*b/a))*d^3/(a*f^4*g^4*n^4*log(F)^4) + 1
/4*(d^3*f^4*g^4*n^4*x^4*log(F)^4 + 4*c*d^2*f^4*g^4*n^4*x^3*log(F)^4 + 6*c^2*d*f^4*g^4*n^4*x^2*log(F)^4)/(a*f^4
*g^4*n^4*log(F)^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (193) = 386\).
time = 0.37, size = 445, normalized size = 2.32 \begin {gather*} -\frac {4 \, {\left (c^{3} f^{3} g^{3} n^{3} - 3 \, c^{2} d f^{2} g^{3} n^{3} e + 3 \, c d^{2} f g^{3} n^{3} e^{2} - d^{3} g^{3} n^{3} e^{3}\right )} \log \left (F^{f g n x + g n e} b + a\right ) \log \left (F\right )^{3} - {\left (d^{3} f^{4} g^{4} n^{4} x^{4} + 4 \, c d^{2} f^{4} g^{4} n^{4} x^{3} + 6 \, c^{2} d f^{4} g^{4} n^{4} x^{2} + 4 \, c^{3} f^{4} g^{4} n^{4} x\right )} \log \left (F\right )^{4} + 4 \, {\left (d^{3} f^{3} g^{3} n^{3} x^{3} + 3 \, c d^{2} f^{3} g^{3} n^{3} x^{2} + 3 \, c^{2} d f^{3} g^{3} n^{3} x + 3 \, c^{2} d f^{2} g^{3} n^{3} e - 3 \, c d^{2} f g^{3} n^{3} e^{2} + d^{3} g^{3} n^{3} e^{3}\right )} \log \left (F\right )^{3} \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right ) + 12 \, {\left (d^{3} f^{2} g^{2} n^{2} x^{2} + 2 \, c d^{2} f^{2} g^{2} n^{2} x + c^{2} d f^{2} g^{2} n^{2}\right )} {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right ) \log \left (F\right )^{2} + 24 \, d^{3} {\rm polylog}\left (4, -\frac {F^{f g n x + g n e} b}{a}\right ) - 24 \, {\left (d^{3} f g n x + c d^{2} f g n\right )} \log \left (F\right ) {\rm polylog}\left (3, -\frac {F^{f g n x + g n e} b}{a}\right )}{4 \, a f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(c^3*f^3*g^3*n^3 - 3*c^2*d*f^2*g^3*n^3*e + 3*c*d^2*f*g^3*n^3*e^2 - d^3*g^3*n^3*e^3)*log(F^(f*g*n*x + g
*n*e)*b + a)*log(F)^3 - (d^3*f^4*g^4*n^4*x^4 + 4*c*d^2*f^4*g^4*n^4*x^3 + 6*c^2*d*f^4*g^4*n^4*x^2 + 4*c^3*f^4*g
^4*n^4*x)*log(F)^4 + 4*(d^3*f^3*g^3*n^3*x^3 + 3*c*d^2*f^3*g^3*n^3*x^2 + 3*c^2*d*f^3*g^3*n^3*x + 3*c^2*d*f^2*g^
3*n^3*e - 3*c*d^2*f*g^3*n^3*e^2 + d^3*g^3*n^3*e^3)*log(F)^3*log((F^(f*g*n*x + g*n*e)*b + a)/a) + 12*(d^3*f^2*g
^2*n^2*x^2 + 2*c*d^2*f^2*g^2*n^2*x + c^2*d*f^2*g^2*n^2)*dilog(-(F^(f*g*n*x + g*n*e)*b + a)/a + 1)*log(F)^2 + 2
4*d^3*polylog(4, -F^(f*g*n*x + g*n*e)*b/a) - 24*(d^3*f*g*n*x + c*d^2*f*g*n)*log(F)*polylog(3, -F^(f*g*n*x + g*
n*e)*b/a))/(a*f^4*g^4*n^4*log(F)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{3}}{a + b \left (F^{e g} F^{f g x}\right )^{n}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+b*(F**(g*(f*x+e)))**n),x)

[Out]

Integral((c + d*x)**3/(a + b*(F**(e*g)*F**(f*g*x))**n), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^3/((F^((f*x + e)*g))^n*b + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n),x)

[Out]

int((c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n), x)

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