\(\int F^{f (a+b \log ^2(c (d+e x)^n))} (d g+e g x) \, dx\) [589]

Optimal result

Integrand size = 29, antiderivative size = 115 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x) \, dx=\frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} g \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]

[Out]

1/2*F^(a*f)*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/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)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2308, 2266, 2235} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x) \, dx=\frac {\sqrt {\pi } g F^{a f} (d+e x)^2 e^{-\frac {1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]

[In]

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

[Out]

(F^(a*f)*g*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*E^(1/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (c (d+e x)^n\right )^{-2/n} (d g+e g x)^2\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 g n} \\ & = \frac {\left (e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} \left (c (d+e x)^n\right )^{-2/n} (d g+e g x)^2\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 g n} \\ & = \frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} g \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 \sqrt {f} n \sqrt {\log (F)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x) \, dx=\frac {e^{-\frac {1}{b f n^2 \log (F)}} F^{a f} g \sqrt {\pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {1+b f n \log (F) \log \left (c (d+e x)^n\right )}{\sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]

[In]

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

[Out]

(F^(a*f)*g*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*E^(1/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x) \, dx=-\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \left (F\right )} g \operatorname {erf}\left (\frac {{\left (b f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b f n \log \left (F\right ) \log \left (c\right ) + 1\right )} \sqrt {-b f n^{2} \log \left (F\right )}}{b f n^{2} \log \left (F\right )}\right ) e^{\left (\frac {a b f^{2} n^{2} \log \left (F\right )^{2} - 2 \, b f n \log \left (F\right ) \log \left (c\right ) - 1}{b f n^{2} \log \left (F\right )}\right )}}{2 \, e n} \]

[In]

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

[Out]

-1/2*sqrt(pi)*sqrt(-b*f*n^2*log(F))*g*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*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (105) = 210\).

Time = 19.43 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.10 \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (d g+e g x) \, dx=\begin {cases} - \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b d^{2} f g n^{2} \log {\left (F \right )}}{2 e} - \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b d^{2} f g n \log {\left (F \right )} \log {\left (c \left (d + e x\right )^{n} \right )}}{2 e} + \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b d f g n^{2} x \log {\left (F \right )}}{2} - F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b d f g n x \log {\left (F \right )} \log {\left (c \left (d + e x\right )^{n} \right )} + \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b e f g n^{2} x^{2} \log {\left (F \right )}}{4} - \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} b e f g n x^{2} \log {\left (F \right )} \log {\left (c \left (d + e x\right )^{n} \right )}}{2} + \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} d^{2} g}{2 e} + F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} d g x + \frac {F^{a f + b f \log {\left (c \left (d + e x\right )^{n} \right )}^{2}} e g x^{2}}{2} & \text {for}\: e \neq 0 \\F^{f \left (a + b \log {\left (c d^{n} \right )}^{2}\right )} d g x & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d**2*f*g*n**2*log(F)/(2*e) - F**(a*f + b*f*log(c*(d + e*x)
**n)**2)*b*d**2*f*g*n*log(F)*log(c*(d + e*x)**n)/(2*e) + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g*n**2*x*
log(F)/2 - F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*d*f*g*n*x*log(F)*log(c*(d + e*x)**n) + F**(a*f + b*f*log(c*
(d + e*x)**n)**2)*b*e*f*g*n**2*x**2*log(F)/4 - F**(a*f + b*f*log(c*(d + e*x)**n)**2)*b*e*f*g*n*x**2*log(F)*log
(c*(d + e*x)**n)/2 + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*d**2*g/(2*e) + F**(a*f + b*f*log(c*(d + e*x)**n)**2
)*d*g*x + F**(a*f + b*f*log(c*(d + e*x)**n)**2)*e*g*x**2/2, Ne(e, 0)), (F**(f*(a + b*log(c*d**n)**2))*d*g*x, T
rue))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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