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

Optimal. Leaf size=56 \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{1}{n},-b \log (F),\frac{1}{n}+1,-\frac{d x^n}{c}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0396952, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{1}{n},-b \log (F),\frac{1}{n}+1,-\frac{d x^n}{c}\right ) \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*Log[c + d*x^n]),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0330856, size = 0, normalized size = 0. \[ \int F^{a+b \log \left (c+d x^n\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c+d*x^n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(F^(b*log(d*x^n + c) + a), 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)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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