3.363 \(\int F^{a+b (c+d x)^n} \, dx\)

Optimal. Leaf size=50 \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

[Out]

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

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Rubi [A]  time = 0.0196655, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.27071, size = 46, normalized size = 0.92 \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{n} \log{\left (F \right )}\right )^{- \frac{1}{n}} \left (c + d x\right ) \Gamma{\left (\frac{1}{n},- b \left (c + d x\right )^{n} \log{\left (F \right )} \right )}}{d n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-F**a*(-b*(c + d*x)**n*log(F))**(-1/n)*(c + d*x)*Gamma(1/n, -b*(c + d*x)**n*log(
F))/(d*n)

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Mathematica [A]  time = 0.0251039, size = 50, normalized size = 1. \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{F}^{a+b \left ( dx+c \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^((d*x + c)^n*b + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(F^((d*x + c)^n*b + a), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(F^((d*x + c)^n*b + a), x)