\(\int F^{f (a+b \log ^2(c (d+e x)^n))} (g+h x)^2 \, dx\) [596]

Optimal result

Integrand size = 28, antiderivative size = 372 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h) \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \]

[Out]

F^(a*f)*h*(-d*h+e*g)*(e*x+d)^2*erfi((1+b*f*n*ln(F)*ln(c*(e*x+d)^n))/n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^
3/exp(1/b/f/n^2/ln(F))/n/((c*(e*x+d)^n)^(2/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2)+1/2*F^(a*f)*(-d*h+e*g)^2*(e*x+d)*er
fi(1/2*(1+2*b*f*n*ln(F)*ln(c*(e*x+d)^n))/n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^3/exp(1/4/b/f/n^2/ln(F))/n/
((c*(e*x+d)^n)^(1/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2)+1/2*F^(a*f)*h^2*(e*x+d)^3*erfi(1/2*(3+2*b*f*n*ln(F)*ln(c*(e*
x+d)^n))/n/b^(1/2)/f^(1/2)/ln(F)^(1/2))*Pi^(1/2)/e^3/exp(9/4/b/f/n^2/ln(F))/n/((c*(e*x+d)^n)^(3/n))/b^(1/2)/f^
(1/2)/ln(F)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2309, 2307, 2266, 2235, 2308} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\frac {\sqrt {\pi } h F^{a f} (d+e x)^2 (e g-d h) e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } F^{a f} (d+e x) (e g-d h)^2 e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^2 F^{a f} (d+e x)^3 e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \]

[In]

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

[Out]

(F^(a*f)*h*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt
[Log[F]])])/(Sqrt[b]*e^3*E^(1/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]]) + (F^(a*f)*(e*g
- d*h)^2*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/
(2*Sqrt[b]*e^3*E^(1/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^n^(-1)*Sqrt[Log[F]]) + (F^(a*f)*h^2*Sqrt[Pi]
*(d + e*x)^3*Erfi[(3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(2*Sqrt[b]*e^3*
E^(9/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F]])

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2307

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.)), x_Symbol] :> Dist[(d + e*x)/(e*n*(c*
(d + e*x)^n)^(1/n)), Subst[Int[E^(a*f*Log[F] + x/n + b*f*Log[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[
{F, a, b, c, d, e, f, n}, x]

Rule 2308

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)), Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Lo
g[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]

Rule 2309

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[1/e^(m + 1), Subst[Int[ExpandIntegrand[F^(f*(a + b*Log[c*x^n]^2)), (e*g - d*h + h*x)^m, x], x], x, d
+ e*x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (e^2 F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g^2 \left (1+\frac {d h (-2 e g+d h)}{e^2 g^2}\right )+2 e F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g h \left (1-\frac {d h}{e g}\right ) x+F^{f \left (a+b \log ^2\left (c x^n\right )\right )} h^2 x^2\right ) \, dx,x,d+e x\right )}{e^3} \\ & = \frac {h^2 \text {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} x^2 \, dx,x,d+e x\right )}{e^3}+\frac {(2 h (e g-d h)) \text {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} x \, dx,x,d+e x\right )}{e^3}+\frac {(e g-d h)^2 \text {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e^3} \\ & = \frac {\left (h^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{\frac {3 x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n}+\frac {\left (2 h (e g-d h) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{\frac {2 x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n}+\frac {\left ((e g-d h)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n} \\ & = \frac {\left (e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{\frac {\left (\frac {3}{n}+2 b f x \log (F)\right )^2}{4 b f \log (F)}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n}+\frac {\left (2 e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{\frac {\left (\frac {2}{n}+2 b f x \log (F)\right )^2}{4 b f \log (F)}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n}+\frac {\left (e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {\left (\frac {1}{n}+2 b f x \log (F)\right )^2}{4 b f \log (F)}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^3 n} \\ & = \frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h) \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{\sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.81 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\frac {e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (-2 e^{\frac {5}{4 b f n^2 \log (F)}} h (-e g+d h) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+e^{\frac {2}{b f n^2 \log (F)}} (e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+h^2 (d+e x)^2 \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )\right )}{2 \sqrt {b} e^3 \sqrt {f} n \sqrt {\log (F)}} \]

[In]

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

[Out]

(F^(a*f)*Sqrt[Pi]*(d + e*x)*(-2*E^(5/(4*b*f*n^2*Log[F]))*h*(-(e*g) + d*h)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erf
i[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + E^(2/(b*f*n^2*Log[F]))*(e*g - d*h)
^2*(c*(d + e*x)^n)^(2/n)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + h^
2*(d + e*x)^2*Erfi[(3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])]))/(2*Sqrt[b]*e^
3*E^(9/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(3/n)*Sqrt[Log[F]])

Maple [F]

\[\int F^{f \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )^{2}\right )} \left (h x +g \right )^{2}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.99 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} h^{2} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 3\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 12 \, b f n \log \left (F\right ) \log \left (c\right ) - 9}{4 \, b f n^{2} \log \left (F\right )}\right )} + \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e^{2} g^{2} - 2 \, d e g h + d^{2} h^{2}\right )} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{2 \, b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \left (F\right )^{2} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b f n^{2} \log \left (F\right )}\right )} + 2 \, \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e g h - d h^{2}\right )} \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \left (F\right )^{2} - 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{b f n^{2} \log \left (F\right )}\right )}}{2 \, e^{3} n} \]

[In]

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

[Out]

-1/2*(sqrt(pi)*sqrt(-b*f*n^2*log(F))*h^2*erf(1/2*(2*b*f*n^2*log(e*x + d)*log(F) + 2*b*f*n*log(F)*log(c) + 3)*s
qrt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^(1/4*(4*a*b*f^2*n^2*log(F)^2 - 12*b*f*n*log(F)*log(c) - 9)/(b*f*n^2*l
og(F))) + sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e^2*g^2 - 2*d*e*g*h + d^2*h^2)*erf(1/2*(2*b*f*n^2*log(e*x + d)*log(F
) + 2*b*f*n*log(F)*log(c) + 1)*sqrt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^(1/4*(4*a*b*f^2*n^2*log(F)^2 - 4*b*f*
n*log(F)*log(c) - 1)/(b*f*n^2*log(F))) + 2*sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e*g*h - d*h^2)*erf((b*f*n^2*log(e*x
+ d)*log(F) + b*f*n*log(F)*log(c) + 1)*sqrt(-b*f*n^2*log(F))/(b*f*n^2*log(F)))*e^((a*b*f^2*n^2*log(F)^2 - 2*b
*f*n*log(F)*log(c) - 1)/(b*f*n^2*log(F))))/(e^3*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (343) = 686\).

Time = 89.20 (sec) , antiderivative size = 1027, normalized size of antiderivative = 2.76 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((-11*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**3*f*h**2*n**2*log(F)/(9*e**3) - 11*F**(a*f + b*f*log
(c*(d + e*x)**n)**2)*b*d**3*f*h**2*n*log(F)*log(c*(d + e*x)**n)/(9*e**3) + 3*F**(a*f + b*f*log(c*(d + e*x)**n)
**2)*b*d**2*f*g*h*n**2*log(F)/e**2 + 3*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**2*f*g*h*n*log(F)*log(c*(d +
e*x)**n)/e**2 + 11*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**2*f*h**2*n**2*x*log(F)/(9*e**2) - 2*F**(a*f + b*
f*log(c*(d + e*x)**n)**2)*b*d**2*f*h**2*n*x*log(F)*log(c*(d + e*x)**n)/(3*e**2) - 2*F**(a*f + b*f*log(c*(d + e
*x)**n)**2)*b*d*f*g**2*n**2*log(F)/e - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g**2*n*log(F)*log(c*(d +
e*x)**n)/e - 3*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g*h*n**2*x*log(F)/e + 2*F**(a*f + b*f*log(c*(d + e*
x)**n)**2)*b*d*f*g*h*n*x*log(F)*log(c*(d + e*x)**n)/e - 5*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*h**2*n**
2*x**2*log(F)/(18*e) + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*h**2*n*x**2*log(F)*log(c*(d + e*x)**n)/(3*e
) + 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g**2*n**2*x*log(F) - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b
*f*g**2*n*x*log(F)*log(c*(d + e*x)**n) + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g*h*n**2*x**2*log(F)/2 - F*
*(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*g*h*n*x**2*log(F)*log(c*(d + e*x)**n) + 2*F**(a*f + b*f*log(c*(d + e*x
)**n)**2)*b*f*h**2*n**2*x**3*log(F)/27 - 2*F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*f*h**2*n*x**3*log(F)*log(c*
(d + e*x)**n)/9 + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*d**3*h**2/(3*e**3) - F**(a*f + b*f*log(c*(d + e*x)**n)
**2)*d**2*g*h/e**2 + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*d*g**2/e + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*g*
*2*x + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*g*h*x**2 + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*h**2*x**3/3, Ne(
e, 0)), (F**(f*(a + b*log(c*d**n)**2))*(g**2*x + g*h*x**2 + h**2*x**3/3), True))

Maxima [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int { {\left (h x + g\right )}^{2} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (g+h\,x\right )}^2 \,d x \]

[In]

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

[Out]

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