\(\int F^{f (a+b \log ^2(c (d+e x)^n))} (g+h x)^3 \, dx\) [595]

Optimal result

Integrand size = 28, antiderivative size = 502 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\frac {3 e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h)^2 \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} h^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^3 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 (e g-d h) \sqrt {\pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}} \]

[Out]

3/2*F^(a*f)*h*(-d*h+e*g)^2*(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^4/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)*h^3*(e*x+d)^4*erf
i((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^4/exp(4/b/f/n^2/ln(F))/n/((c*(e*x+
d)^n)^(4/n))/b^(1/2)/f^(1/2)/ln(F)^(1/2)+1/2*F^(a*f)*(-d*h+e*g)^3*(e*x+d)*erfi(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^4/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)+3/2*F^(a*f)*h^2*(-d*h+e*g)*(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^4/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 = 0.60 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2309, 2307, 2266, 2235, 2308} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\frac {3 \sqrt {\pi } h^2 F^{a f} (d+e x)^3 (e g-d h) e^{-\frac {9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 \sqrt {\pi } h F^{a f} (d+e x)^2 (e g-d h)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } F^{a f} (d+e x) (e g-d h)^3 e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {\sqrt {\pi } h^3 F^{a f} (d+e x)^4 e^{-\frac {4}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+2}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}} \]

[In]

Int[F^(f*(a + b*Log[c*(d + e*x)^n]^2))*(g + h*x)^3,x]

[Out]

(3*F^(a*f)*h*(e*g - d*h)^2*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]])])/(2*Sqrt[b]*e^4*E^(1/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]]) + (F^(a*f)
*h^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(2*Sqr
t[b]*e^4*E^(4/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(4/n)*Sqrt[Log[F]]) + (F^(a*f)*(e*g - d*h)^3*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^4*E
^(1/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^n^(-1)*Sqrt[Log[F]]) + (3*F^(a*f)*h^2*(e*g - d*h)*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^4*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^3 F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g^3 \left (1-\frac {d h \left (3 e^2 g^2-3 d e g h+d^2 h^2\right )}{e^3 g^3}\right )+3 e^2 F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g^2 h \left (1+\frac {d h (-2 e g+d h)}{e^2 g^2}\right ) x+3 e F^{f \left (a+b \log ^2\left (c x^n\right )\right )} g h^2 \left (1-\frac {d h}{e g}\right ) x^2+F^{f \left (a+b \log ^2\left (c x^n\right )\right )} h^3 x^3\right ) \, dx,x,d+e x\right )}{e^4} \\ & = \frac {h^3 \text {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} x^3 \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 h^2 (e g-d h)\right ) \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^4}+\frac {\left (3 h (e g-d h)^2\right ) \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^4}+\frac {(e g-d h)^3 \text {Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e^4} \\ & = \frac {\left (h^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int e^{\frac {4 x}{n}+a f \log (F)+b f x^2 \log (F)} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 h^2 (e g-d h) (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^4 n}+\frac {\left (3 h (e g-d h)^2 (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^4 n}+\frac {\left ((e g-d h)^3 (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^4 n} \\ & = \frac {\left (e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} h^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int e^{\frac {\left (\frac {4}{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^4 n}+\frac {\left (3 e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 (e g-d h) (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^4 n}+\frac {\left (3 e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h)^2 (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^4 n}+\frac {\left (e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^3 (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^4 n} \\ & = \frac {3 e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} h (e g-d h)^2 \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 )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} h^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {e^{-\frac {1}{4 b f n^2 \log (F)}} F^{a f} (e g-d h)^3 \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^4 \sqrt {f} n \sqrt {\log (F)}}+\frac {3 e^{-\frac {9}{4 b f n^2 \log (F)}} F^{a f} h^2 (e g-d h) \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^4 \sqrt {f} n \sqrt {\log (F)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.79 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\frac {e^{-\frac {4}{b f n^2 \log (F)}} F^{a f} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (3 e^{\frac {3}{b f n^2 \log (F)}} h (e g-d h)^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+h^3 (d+e x)^3 \text {erfi}\left (\frac {2+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+e^{\frac {7}{4 b f n^2 \log (F)}} (e g-d h) \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (e^{\frac {2}{b f n^2 \log (F)}} (e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {1+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )+3 h^2 (d+e x)^2 \text {erfi}\left (\frac {3+2 b f n \log (F) \log \left (c (d+e x)^n\right )}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )\right )\right )}{2 \sqrt {b} e^4 \sqrt {f} n \sqrt {\log (F)}} \]

[In]

Integrate[F^(f*(a + b*Log[c*(d + e*x)^n]^2))*(g + h*x)^3,x]

[Out]

(F^(a*f)*Sqrt[Pi]*(d + e*x)*(3*E^(3/(b*f*n^2*Log[F]))*h*(e*g - d*h)^2*(d + e*x)*(c*(d + e*x)^n)^(2/n)*Erfi[(1
+ b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + h^3*(d + e*x)^3*Erfi[(2 + b*f*n*Log[F]*
Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + E^(7/(4*b*f*n^2*Log[F]))*(e*g - d*h)*(c*(d + e*x)^n)^n
^(-1)*(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]])] + 3*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^4*E^(4/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(4/n)*Sqrt[Log[
F]])

Maple [F]

\[\int F^{f \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )^{2}\right )} \left (h x +g \right )^{3}d x\]

[In]

int(F^(f*(a+b*ln(c*(e*x+d)^n)^2))*(h*x+g)^3,x)

[Out]

int(F^(f*(a+b*ln(c*(e*x+d)^n)^2))*(h*x+g)^3,x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.02 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} h^{3} \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 ) + 2\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} - 4 \, b f n \log \left (F\right ) \log \left (c\right ) - 4}{b f n^{2} \log \left (F\right )}\right )} + \sqrt {\pi } {\left (e^{3} g^{3} - 3 \, d e^{2} g^{2} h + 3 \, d^{2} e g h^{2} - d^{3} h^{3}\right )} \sqrt {-b f n^{2} \log \left (F\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 )} + 3 \, \sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} {\left (e g h^{2} - d h^{3}\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 ) + 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 )} + 3 \, \sqrt {\pi } {\left (e^{2} g^{2} h - 2 \, d e g h^{2} + d^{2} h^{3}\right )} \sqrt {-b f n^{2} \log \left (F\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^{4} n} \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n)^2))*(h*x+g)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\text {Timed out} \]

[In]

integrate(F**(f*(a+b*ln(c*(e*x+d)**n)**2))*(h*x+g)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n)^2))*(h*x+g)^3,x, algorithm="maxima")

[Out]

integrate((h*x + g)^3*F^((b*log((e*x + d)^n*c)^2 + a)*f), x)

Giac [F]

\[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f} \,d x } \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n)^2))*(h*x+g)^3,x, algorithm="giac")

[Out]

integrate((h*x + g)^3*F^((b*log((e*x + d)^n*c)^2 + a)*f), x)

Mupad [F(-1)]

Timed out. \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,\left (b\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2+a\right )}\,{\left (g+h\,x\right )}^3 \,d x \]

[In]

int(F^(f*(a + b*log(c*(d + e*x)^n)^2))*(g + h*x)^3,x)

[Out]

int(exp(f*log(F)*(a + b*log(c*(d + e*x)^n)^2))*(g + h*x)^3, x)