3.580 \(\int F^{a+b \log \left (c+d x^n\right )} x \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{2} x^2 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{2}{n},-b \log (F),\frac{n+2}{n},-\frac{d x^n}{c}\right ) \]

[Out]

(F^a*x^2*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[2/n, -(b*Log[F]), (2 + n)/n, -
((d*x^n)/c)])/(2*(1 + (d*x^n)/c)^(b*Log[F]))

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Rubi [A]  time = 0.0673641, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{2} x^2 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{2}{n},-b \log (F),\frac{n+2}{n},-\frac{d x^n}{c}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(F^a*x^2*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[2/n, -(b*Log[F]), (2 + n)/n, -
((d*x^n)/c)])/(2*(1 + (d*x^n)/c)^(b*Log[F]))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{a + b \log{\left (c + d x^{n} \right )}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*ln(c+d*x**n))*x,x)

[Out]

Integral(F**(a + b*log(c + d*x**n))*x, x)

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Mathematica [A]  time = 0.239701, size = 0, normalized size = 0. \[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c+d*x^n))*x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(d*x^n + c) + a)*x,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (F^{b \log \left (d x^{n} + c\right ) + a} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(d*x^n + c) + a)*x,x, algorithm="fricas")

[Out]

integral(F^(b*log(d*x^n + c) + a)*x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*ln(c+d*x**n))*x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int F^{b \log \left (d x^{n} + c\right ) + a} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(d*x^n + c) + a)*x,x, algorithm="giac")

[Out]

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