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

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]

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

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Rubi [A]  time = 0.09, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 118, normalized size = 1.00 \[ \frac {\sqrt {\pi } F^{a f} (d+e x) e^{-\frac {1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt {b} \sqrt {f} n \sqrt {\log (F)}}\right )}{2 \sqrt {b} e \sqrt {f} n \sqrt {\log (F)}} \]

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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fricas [A]  time = 0.42, size = 116, normalized size = 0.98 \[ -\frac {\sqrt {\pi } \sqrt {-b f n^{2} \log \relax (F)} \operatorname {erf}\left (\frac {{\left (2 \, b f n^{2} \log \left (e x + d\right ) \log \relax (F) + 2 \, b f n \log \relax (F) \log \relax (c) + 1\right )} \sqrt {-b f n^{2} \log \relax (F)}}{2 \, b f n^{2} \log \relax (F)}\right ) e^{\left (\frac {4 \, a b f^{2} n^{2} \log \relax (F)^{2} - 4 \, b f n \log \relax (F) \log \relax (c) - 1}{4 \, b f n^{2} \log \relax (F)}\right )}}{2 \, e n} \]

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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giac [A]  time = 0.40, size = 101, normalized size = 0.86 \[ -\frac {\sqrt {\pi } F^{a f} \operatorname {erf}\left (-\sqrt {-b f \log \relax (F)} n \log \left (x e + d\right ) - \sqrt {-b f \log \relax (F)} \log \relax (c) - \frac {\sqrt {-b f \log \relax (F)}}{2 \, b f n \log \relax (F)}\right ) e^{\left (-\frac {1}{4 \, b f n^{2} \log \relax (F)} - 1\right )}}{2 \, \sqrt {-b f \log \relax (F)} c^{\left (\frac {1}{n}\right )} n} \]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a\right )} f}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{b\,f\,{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2}\,F^{a\,f} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 29.05, size = 532, normalized size = 4.51 \[ \begin {cases} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n^{2} \log {\relax (F )} \log {\left (d + e x \right )}}{e} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n^{2} \log {\relax (F )}}{e} - \frac {2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b d f n \log {\relax (F )} \log {\relax (c )}}{e} - 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n^{2} x \log {\relax (F )} \log {\left (d + e x \right )} + 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n^{2} x \log {\relax (F )} - 2 F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} b f n x \log {\relax (F )} \log {\relax (c )} + \frac {F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} d}{e} + F^{a f} F^{b f \log {\relax (c )}^{2}} F^{b f n^{2} \log {\left (d + e x \right )}^{2}} F^{2 b f n \log {\relax (c )} \log {\left (d + e x \right )}} x & \text {for}\: e \neq 0 \\F^{f \left (a + b \log {\left (c d^{n} \right )}^{2}\right )} x & \text {otherwise} \end {cases} \]

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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