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

Optimal. Leaf size=118 \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]

[Out]

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

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Rubi [A]  time = 0.126591, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2275, 2234, 2204} \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2275

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

Rule 2234

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 2204

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]

Rubi steps

\begin{align*} \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int F^{f \left (a+b \log ^2\left (c x^n\right )\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{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 n}\\ &=\frac{\left (e^{-\frac{1}{4 b f n^2 \log (F)}} F^{a f} (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{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 n}\\ &=\frac{e^{-\frac{1}{4 b f n^2 \log (F)}} F^{a f} \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 \sqrt{f} n \sqrt{\log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.095201, size = 118, normalized size = 1. \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{F}^{f \left ( a+b \left ( \ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.02131, size = 309, normalized size = 2.62 \begin{align*} -\frac{\sqrt{\pi } \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 )}}{2 \, e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*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))
)/(e*n)

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Sympy [A]  time = 107.124, size = 532, normalized size = 4.51 \begin{align*} \begin{cases} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n^{2} \log{\left (F \right )} \log{\left (d + e x \right )}}{e} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n^{2} \log{\left (F \right )}}{e} - \frac{2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b d f n \log{\left (F \right )} \log{\left (c \right )}}{e} - 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n^{2} x \log{\left (F \right )} \log{\left (d + e x \right )} + 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n^{2} x \log{\left (F \right )} - 2 F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} b f n x \log{\left (F \right )} \log{\left (c \right )} + \frac{F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} d}{e} + F^{a f} F^{b f \log{\left (c \right )}^{2}} F^{b f n^{2} \log{\left (d + e x \right )}^{2}} F^{2 b f n \log{\left (c \right )} \log{\left (d + e x \right )}} x & \text{for}\: e \neq 0 \\F^{f \left (a + b \log{\left (c d^{n} \right )}^{2}\right )} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d + e*x)**2)*F**(2*b*f*n*log(c)*log(d + e*x))*b*d*f
*n**2*log(F)*log(d + e*x)/e - 2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d + e*x)**2)*F**(2*b*f*n*log(c)*l
og(d + e*x))*b*d*f*n**2*log(F)/e - 2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d + e*x)**2)*F**(2*b*f*n*log
(c)*log(d + e*x))*b*d*f*n*log(F)*log(c)/e - 2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d + e*x)**2)*F**(2*
b*f*n*log(c)*log(d + e*x))*b*f*n**2*x*log(F)*log(d + e*x) + 2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d +
e*x)**2)*F**(2*b*f*n*log(c)*log(d + e*x))*b*f*n**2*x*log(F) - 2*F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(
d + e*x)**2)*F**(2*b*f*n*log(c)*log(d + e*x))*b*f*n*x*log(F)*log(c) + F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2
*log(d + e*x)**2)*F**(2*b*f*n*log(c)*log(d + e*x))*d/e + F**(a*f)*F**(b*f*log(c)**2)*F**(b*f*n**2*log(d + e*x)
**2)*F**(2*b*f*n*log(c)*log(d + e*x))*x, Ne(e, 0)), (F**(f*(a + b*log(c*d**n)**2))*x, True))

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Giac [A]  time = 1.29082, size = 136, normalized size = 1.15 \begin{align*} -\frac{\sqrt{\pi } F^{a f} \operatorname{erf}\left (-\sqrt{-b f \log \left (F\right )} n \log \left (x e + d\right ) - \sqrt{-b f \log \left (F\right )} \log \left (c\right ) - \frac{\sqrt{-b f \log \left (F\right )}}{2 \, b f n \log \left (F\right )}\right ) e^{\left (-\frac{1}{4 \, b f n^{2} \log \left (F\right )} - 1\right )}}{2 \, \sqrt{-b f \log \left (F\right )} c^{\left (\frac{1}{n}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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