3.250 \(\int f^{c (a+b x)^n} \, dx\)

Optimal. Leaf size=47 \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b n} \]

[Out]

-(((a + b*x)*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b*n*(-(c*(a + b*x)^n*Log[f
]))^n^(-1)))

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Rubi [A]  time = 0.017441, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]  Int[f^(c*(a + b*x)^n),x]

[Out]

-(((a + b*x)*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b*n*(-(c*(a + b*x)^n*Log[f
]))^n^(-1)))

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Rubi in Sympy [A]  time = 2.03448, size = 42, normalized size = 0.89 \[ - \frac{\left (- c \left (a + b x\right )^{n} \log{\left (f \right )}\right )^{- \frac{1}{n}} \left (a + b x\right ) \Gamma{\left (\frac{1}{n},- c \left (a + b x\right )^{n} \log{\left (f \right )} \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0112775, size = 47, normalized size = 1. \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c*(a + b*x)^n),x]

[Out]

-(((a + b*x)*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b*n*(-(c*(a + b*x)^n*Log[f
]))^n^(-1)))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{f}^{c \left ( bx+a \right ) ^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^n*c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(f^((b*x + a)^n*c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c*(b*x+a)**n),x)

[Out]

Integral(f**(c*(a + b*x)**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^n*c), x)