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

Optimal. Leaf size=397 \[ \frac{\sqrt{\pi } h^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac{3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \text{Erfi}\left (\frac{2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{3}{n}}{2 b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e^3 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } h (d+e x)^2 (e g-d h) \left (c (d+e x)^n\right )^{-2/n} e^{-\frac{2 a b f n \log (F)+1}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{a b f \log (F)+b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{b \sqrt{f} \sqrt{\log (F)}}\right )}{b e^3 \sqrt{f} n \sqrt{\log (F)}}+\frac{\sqrt{\pi } (d+e x) (e g-d h)^2 \left (c (d+e x)^n\right )^{-1/n} e^{-\frac{4 a b f n \log (F)+1}{4 b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )+\frac{1}{n}}{2 b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e^3 \sqrt{f} n \sqrt{\log (F)}} \]

[Out]

(h*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^2*Erfi[(n^(-1) + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*
Sqrt[Log[F]])])/(b*e^3*E^((1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^(2/n)*Sqrt[Log[
F]]) + ((e*g - d*h)^2*Sqrt[Pi]*(d + e*x)*Erfi[(n^(-1) + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2
*b*Sqrt[f]*Sqrt[Log[F]])])/(2*b*e^3*E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*Sqrt[f]*n*(c*(d + e*x)^n)^
n^(-1)*Sqrt[Log[F]]) + (h^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n
])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(2*b*e^3*E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*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 = 0.367427, 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 = {} \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^2 \, dx \]

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]

(g^2*Sqrt[Pi]*(d + e*x)*Erfi[(n^(-1) + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[L
og[F]])])/(2*b*e*E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*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 \left (c (d+e x)^n\right )\right )^2} (g+h x)^2 \, dx &=\int \left (F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} g^2+2 F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} g h x+F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} h^2 x^2\right ) \, dx\\ &=g^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx+(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{f \left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e}+(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{a^2 f+2 a b f \log \left (c x^n\right )+b^2 f \log ^2\left (c x^n\right )} \, dx,x,d+e x\right )}{e}+(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx\\ &=\frac{g^2 \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} \left (c x^n\right )^{2 a b f \log (F)} \, dx,x,d+e x\right )}{e}+(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx\\ &=(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx+\frac{\left (g^2 (d+e x)^{-2 a b f n \log (F)} \left (c (d+e x)^n\right )^{2 a b f \log (F)}\right ) \operatorname{Subst}\left (\int F^{a^2 f+b^2 f \log ^2\left (c x^n\right )} x^{2 a b f n \log (F)} \, dx,x,d+e x\right )}{e}\\ &=(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx+\frac{\left (g^2 (d+e x) \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{1+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (a^2 f \log (F)+b^2 f x^2 \log (F)+\frac{x (1+2 a b f n \log (F))}{n}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx+\frac{\left (\exp \left (a^2 f \log (F)-\frac{(1+2 a b f n \log (F))^2}{4 b^2 f n^2 \log (F)}\right ) g^2 (d+e x) \left (c (d+e x)^n\right )^{2 a b f \log (F)-\frac{1+2 a b f n \log (F)}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (\frac{\left (2 b^2 f x \log (F)+\frac{1+2 a b f n \log (F)}{n}\right )^2}{4 b^2 f \log (F)}\right ) \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{e^{-\frac{1+4 a b f n \log (F)}{4 b^2 f n^2 \log (F)}} g^2 \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\frac{1}{n}+2 a b f \log (F)+2 b^2 f \log (F) \log \left (c (d+e x)^n\right )}{2 b \sqrt{f} \sqrt{\log (F)}}\right )}{2 b e \sqrt{f} n \sqrt{\log (F)}}+(2 g h) \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x \, dx+h^2 \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} x^2 \, dx\\ \end{align*}

Mathematica [A]  time = 1.0468, size = 331, normalized size = 0.83 \[ \frac{\sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \exp \left (-\frac{3 (4 a b f n \log (F)+3)}{4 b^2 f n^2 \log (F)}\right ) \left ((e g-d h)^2 \left (c (d+e x)^n\right )^{2/n} e^{\frac{2 a b f n \log (F)+2}{b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )+1}{2 b \sqrt{f} n \sqrt{\log (F)}}\right )-2 h (d+e x) (d h-e g) \left (c (d+e x)^n\right )^{\frac{1}{n}} e^{\frac{4 a b f n \log (F)+5}{4 b^2 f n^2 \log (F)}} \text{Erfi}\left (\frac{b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )+1}{b \sqrt{f} n \sqrt{\log (F)}}\right )+h^2 (d+e x)^2 \text{Erfi}\left (\frac{2 b f n \log (F) \left (a+b \log \left (c (d+e x)^n\right )\right )+3}{2 b \sqrt{f} n \sqrt{\log (F)}}\right )\right )}{2 b e^3 \sqrt{f} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(d + e*x)*(-2*E^((5 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*h*(-(e*g) + d*h)*(d + e*x)*(c*(d + e*x
)^n)^n^(-1)*Erfi[(1 + b*f*n*Log[F]*(a + b*Log[c*(d + e*x)^n]))/(b*Sqrt[f]*n*Sqrt[Log[F]])] + E^((2 + 2*a*b*f*n
*Log[F])/(b^2*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]*(a + b*Log[c*(d + e*
x)^n]))/(2*b*Sqrt[f]*n*Sqrt[Log[F]])] + h^2*(d + e*x)^2*Erfi[(3 + 2*b*f*n*Log[F]*(a + b*Log[c*(d + e*x)^n]))/(
2*b*Sqrt[f]*n*Sqrt[Log[F]])]))/(2*b*e^3*E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*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 = 0.483, size = 0, normalized size = 0. \begin{align*} \int{F}^{f \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}} \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 ) + a\right )}^{2} 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) + a)^2*f), x)

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Fricas [A]  time = 1.12098, size = 1076, normalized size = 2.71 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b^{2} f n^{2} \log \left (F\right )} h^{2} \operatorname{erf}\left (\frac{{\left (2 \, b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 3\right )} \sqrt{-b^{2} f n^{2} \log \left (F\right )}}{2 \, b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac{3 \,{\left (4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) + 3\right )}}{4 \, b^{2} f n^{2} \log \left (F\right )}\right )} + \sqrt{\pi } \sqrt{-b^{2} 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^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + 2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 1\right )} \sqrt{-b^{2} f n^{2} \log \left (F\right )}}{2 \, b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac{4 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 4 \, a b f n \log \left (F\right ) + 1}{4 \, b^{2} f n^{2} \log \left (F\right )}\right )} + 2 \, \sqrt{\pi } \sqrt{-b^{2} f n^{2} \log \left (F\right )}{\left (e g h - d h^{2}\right )} \operatorname{erf}\left (\frac{{\left (b^{2} f n^{2} \log \left (e x + d\right ) \log \left (F\right ) + b^{2} f n \log \left (F\right ) \log \left (c\right ) + a b f n \log \left (F\right ) + 1\right )} \sqrt{-b^{2} f n^{2} \log \left (F\right )}}{b^{2} f n^{2} \log \left (F\right )}\right ) e^{\left (-\frac{2 \, b^{2} f n \log \left (F\right ) \log \left (c\right ) + 2 \, a b f n \log \left (F\right ) + 1}{b^{2} f n^{2} \log \left (F\right )}\right )}}{2 \, b 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^2*f*n^2*log(F))*h^2*erf(1/2*(2*b^2*f*n^2*log(e*x + d)*log(F) + 2*b^2*f*n*log(F)*log(c)
+ 2*a*b*f*n*log(F) + 3)*sqrt(-b^2*f*n^2*log(F))/(b^2*f*n^2*log(F)))*e^(-3/4*(4*b^2*f*n*log(F)*log(c) + 4*a*b*f
*n*log(F) + 3)/(b^2*f*n^2*log(F))) + sqrt(pi)*sqrt(-b^2*f*n^2*log(F))*(e^2*g^2 - 2*d*e*g*h + d^2*h^2)*erf(1/2*
(2*b^2*f*n^2*log(e*x + d)*log(F) + 2*b^2*f*n*log(F)*log(c) + 2*a*b*f*n*log(F) + 1)*sqrt(-b^2*f*n^2*log(F))/(b^
2*f*n^2*log(F)))*e^(-1/4*(4*b^2*f*n*log(F)*log(c) + 4*a*b*f*n*log(F) + 1)/(b^2*f*n^2*log(F))) + 2*sqrt(pi)*sqr
t(-b^2*f*n^2*log(F))*(e*g*h - d*h^2)*erf((b^2*f*n^2*log(e*x + d)*log(F) + b^2*f*n*log(F)*log(c) + a*b*f*n*log(
F) + 1)*sqrt(-b^2*f*n^2*log(F))/(b^2*f*n^2*log(F)))*e^(-(2*b^2*f*n*log(F)*log(c) + 2*a*b*f*n*log(F) + 1)/(b^2*
f*n^2*log(F))))/(b*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 ) + a\right )}^{2} 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) + a)^2*f), x)