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

Optimal. Leaf size=502 \[ \frac{3 \sqrt{\pi } h^2 F^{a f} (d+e x)^3 (e g-d h) e^{-\frac{9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^4 \sqrt{f} n \sqrt{\log (F)}}+\frac{3 \sqrt{\pi } h F^{a f} (d+e x)^2 (e g-d h)^2 e^{-\frac{1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^4 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } F^{a f} (d+e x) (e g-d h)^3 e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^4 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } h^3 F^{a f} (d+e x)^4 e^{-\frac{4}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-4/n} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+2}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^4 \sqrt{f} n \sqrt{\log (F)}} \]

[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]])

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Rubi [F]  time = 0.376368, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.55108, size = 396, normalized size = 0.79 \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{4}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-4/n} \left ((e g-d h) e^{\frac{7}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \left ((e g-d h)^2 e^{\frac{2}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )+3 h^2 (d+e x)^2 \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )\right )+3 h (d+e x) (e g-d h)^2 e^{\frac{3}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )+h^3 (d+e x)^3 \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+2}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )\right )}{2 \sqrt{b} e^4 \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[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]])

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Maple [F]  time = 0.552, 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 ) } \left ( hx+g \right ) ^{3}\, 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))*(h*x+g)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \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))*(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)

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Fricas [A]  time = 1.093, size = 1319, normalized size = 2.63 \begin{align*} -\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} \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))*(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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))*(h*x+g)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \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))*(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)