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

Optimal. Leaf size=372 \[ \frac{\sqrt{\pi } h F^{a f} (d+e x)^2 (e g-d h) e^{-\frac{1}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{\sqrt{b} e^3 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } F^{a f} (d+e x) (e g-d h)^2 e^{-\frac{1}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^3 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } h^2 F^{a f} (d+e x)^3 e^{-\frac{9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} e^3 \sqrt{f} n \sqrt{\log (F)}} \]

[Out]

(F^(a*f)*h*(e*g - d*h)*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]])])/(Sqrt[b]*e^3*E^(1/(b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[F]]) + (F^(a*f)*(e*g
- d*h)^2*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^3*E^(1/(4*b*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^n^(-1)*Sqrt[Log[F]]) + (F^(a*f)*h^2*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^3*
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 = 0.306164, 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 = {} \[ \int F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )} (g+h x)^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.703055, size = 303, normalized size = 0.81 \[ \frac{\sqrt{\pi } F^{a f} (d+e x) e^{-\frac{9}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{-3/n} \left ((e g-d h)^2 e^{\frac{2}{b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{2/n} \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+1}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )-2 h (d+e x) (d h-e g) e^{\frac{5}{4 b f n^2 \log (F)}} \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{b f n \log (F) \log \left (c (d+e x)^n\right )+1}{\sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )+h^2 (d+e x)^2 \text{Erfi}\left (\frac{2 b f n \log (F) \log \left (c (d+e x)^n\right )+3}{2 \sqrt{b} \sqrt{f} n \sqrt{\log (F)}}\right )\right )}{2 \sqrt{b} e^3 \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a*f)*Sqrt[Pi]*(d + e*x)*(-2*E^(5/(4*b*f*n^2*Log[F]))*h*(-(e*g) + d*h)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erf
i[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])] + 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]])] + 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^
3*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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Maple [F]  time = 0.458, 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 ) ^{2}\, 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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (h x + g\right )}^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.03795, size = 961, normalized size = 2.58 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b f n^{2} \log \left (F\right )} h^{2} \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 )} + \sqrt{\pi } \sqrt{-b f n^{2} \log \left (F\right )}{\left (e^{2} g^{2} - 2 \, d e g h + d^{2} h^{2}\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 \, \sqrt{\pi } \sqrt{-b f n^{2} \log \left (F\right )}{\left (e g h - d h^{2}\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^{3} 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)^2,x, algorithm="fricas")

[Out]

-1/2*(sqrt(pi)*sqrt(-b*f*n^2*log(F))*h^2*erf(1/2*(2*b*f*n^2*log(e*x + d)*log(F) + 2*b*f*n*log(F)*log(c) + 3)*s
qrt(-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*l
og(F))) + sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e^2*g^2 - 2*d*e*g*h + d^2*h^2)*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))) + 2*sqrt(pi)*sqrt(-b*f*n^2*log(F))*(e*g*h - d*h^2)*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^3*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)**2,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 )}^{2} 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)^2,x, algorithm="giac")

[Out]

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