3.53 \(\int \frac{(c+d x)^2}{(a+b (F^{g (e+f x)})^n)^2} \, dx\)

Optimal. Leaf size=294 \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^2}{a^2 f g n \log (F)}+\frac{(c+d x)^3}{3 a^2 d}+\frac{(c+d x)^2}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.671343, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ -\frac{2 d (c+d x) \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}+\frac{2 d^2 \text{PolyLog}\left (2,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d^2 \text{PolyLog}\left (3,-\frac{b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}+\frac{2 d (c+d x) \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac{(c+d x)^2 \log \left (\frac{b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a^2 f g n \log (F)}-\frac{(c+d x)^2}{a^2 f g n \log (F)}+\frac{(c+d x)^3}{3 a^2 d}+\frac{(c+d x)^2}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

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

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

Mathematica [F]  time = 1.12444, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^2,x]

[Out]

Integrate[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^2, x]

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Maple [B]  time = 0.085, size = 1754, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^2,x)

[Out]

-2/3/g^3/f^3/ln(F)^3/a^2*d^2*ln(F^(g*(f*x+e)))^3+2/n/g/f/ln(F)/a^2*c*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F
^(g*(f*x+e))))))*x-2/n/g^2/f^2/ln(F)^2/a^2*c*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g
*(f*x+e)))-2/n/g/f/ln(F)/a^2*c*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+2/n/g^2/f^2/ln(
F)^2/a^2*c*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))-2/n/g^2/f^2/ln(F)^2
/a^2*c*d*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+e)))-1/n/g^3/f^3/ln(F)^3/a
^2*d^2*ln(F^(g*(f*x+e)))^2+2/n^3/g^3/f^3/ln(F)^3/a^2*d^2*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*
(f*x+e)))))/a)+2/n^3/g^3/f^3/ln(F)^3/a^2*d^2*polylog(3,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/
a)-1/n/g/f/ln(F)/a^2*c^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+1/n/g/f/ln(F)/a^2*c^2*ln(
F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))+1/g^2/f^2/ln(F)^2/a^2*d^2*ln(F^(g*(f*x+e)))^2*x+1/g^2/f^2
/ln(F)^2/a^2*c*d*ln(F^(g*(f*x+e)))^2+2/n/g^2/f^2/ln(F)^2/a^2*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(
g*(f*x+e))))))*ln(F^(g*(f*x+e)))*x-2/n/g^2/f^2/ln(F)^2/a^2*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x
+e))))))*ln(F^(g*(f*x+e)))*x-2/n/g^2/f^2/ln(F)^2/a^2*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e
)))))/a)*ln(F^(g*(f*x+e)))*x+1/n/g/f/ln(F)/a*(d^2*x^2+2*c*d*x+c^2)/(a+b*(F^(g*(f*x+e)))^n)-2/n^2/g^2/f^2/ln(F)
^2/a^2*c*d*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))-1/n/g/f/ln(F)/a^2*d^2*ln(a+b*F^(n*g*f*x)*ex
p(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x^2-1/n/g^3/f^3/ln(F)^3/a^2*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-
ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))^2+1/n/g/f/ln(F)/a^2*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x
+e))))))*x^2+1/n/g^3/f^3/ln(F)^3/a^2*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+
e)))^2+1/n/g^3/f^3/ln(F)^3/a^2*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*ln(F^(g*(f*x+
e)))^2+2/n^2/g^2/f^2/ln(F)^2/a^2*c*d*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))-2/n^2/g^2/f^2
/ln(F)^2/a^2*d^2*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)*x-2/n^2/g^2/f^2/ln(F)^2/a
^2*c*d*polylog(2,-b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e)))))/a)-2/n^2/g^2/f^2/ln(F)^2/a^2*d^2*ln(F^
(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e))))))*x+2/n^2/g^3/f^3/ln(F)^3/a^2*d^2*ln(F^(n*g*f*x)*exp(-n*(ln(F
)*f*g*x-ln(F^(g*(f*x+e))))))*ln(F^(g*(f*x+e)))+2/n^2/g^2/f^2/ln(F)^2/a^2*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*
f*g*x-ln(F^(g*(f*x+e))))))*x-2/n^2/g^3/f^3/ln(F)^3/a^2*d^2*ln(a+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x
+e))))))*ln(F^(g*(f*x+e)))+2/n^2/g^3/f^3/ln(F)^3/a^2*d^2*ln(1+b*F^(n*g*f*x)*exp(-n*(ln(F)*f*g*x-ln(F^(g*(f*x+e
)))))/a)*ln(F^(g*(f*x+e)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2}{\left (\frac{1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \left (F\right )} + \frac{\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \left (F\right )} - \frac{\log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac{d^{2} x^{2} + 2 \, c d x}{{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} + \int \frac{d^{2} f g n x^{2} \log \left (F\right ) - 2 \, c d + 2 \,{\left (c d f g n \log \left (F\right ) - d^{2}\right )} x}{{\left (F^{f g x}\right )}^{n}{\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^2,x, algorithm="maxima")

[Out]

c^2*(1/(((F^(f*g*x + e*g))^n*a*b*n + a^2*n)*f*g*log(F)) + log(F^(f*g*x + e*g))/(a^2*f*g*log(F)) - log(((F^(f*g
*x + e*g))^n*b + a)/b)/(a^2*f*g*n*log(F))) + (d^2*x^2 + 2*c*d*x)/((F^(f*g*x))^n*(F^(e*g))^n*a*b*f*g*n*log(F) +
a^2*f*g*n*log(F)) + integrate((d^2*f*g*n*x^2*log(F) - 2*c*d + 2*(c*d*f*g*n*log(F) - d^2)*x)/((F^(f*g*x))^n*(F
^(e*g))^n*a*b*f*g*n*log(F) + a^2*f*g*n*log(F)), x)

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Fricas [C]  time = 1.64334, size = 1823, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+b*(F**(g*(f*x+e)))**n)**2,x)

[Out]

(c**2 + 2*c*d*x + d**2*x**2)/(a**2*f*g*n*log(F) + a*b*f*g*n*(F**(g*(e + f*x)))**n*log(F)) + (Integral(-2*c*d/(
a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(-2*d**2*x/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log
(F))), x) + Integral(c**2*f*g*n*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(d**2*f*g*n
*x**2*log(F)/(a + b*exp(e*g*n*log(F))*exp(f*g*n*x*log(F))), x) + Integral(2*c*d*f*g*n*x*log(F)/(a + b*exp(e*g*
n*log(F))*exp(f*g*n*x*log(F))), x))/(a*f*g*n*log(F))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^2/((F^((f*x + e)*g))^n*b + a)^2, x)