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

Optimal. Leaf size=99 \[ \frac{a (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^2 n}-\frac{(a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^2 n} \]

[Out]

-(((a + b*x)^2*Gamma[2/n, -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[
f]))^(2/n))) + (a*(a + b*x)*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*
(a + b*x)^n*Log[f]))^n^(-1))

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Rubi [A]  time = 0.102659, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^2 n}-\frac{(a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[f^(c*(a + b*x)^n)*x,x]

[Out]

-(((a + b*x)^2*Gamma[2/n, -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*(a + b*x)^n*Log[
f]))^(2/n))) + (a*(a + b*x)*Gamma[n^(-1), -(c*(a + b*x)^n*Log[f])])/(b^2*n*(-(c*
(a + b*x)^n*Log[f]))^n^(-1))

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Rubi in Sympy [A]  time = 10.2292, size = 90, normalized size = 0.91 \[ \frac{a \left (- c \left (a + b x\right )^{n} \log{\left (f \right )}\right )^{- \frac{1}{n}} \left (a + b x\right ) \Gamma{\left (\frac{1}{n},- c \left (a + b x\right )^{n} \log{\left (f \right )} \right )}}{b^{2} n} - \frac{\left (- c \left (a + b x\right )^{n} \log{\left (f \right )}\right )^{- \frac{2}{n}} \left (a + b x\right )^{2} \Gamma{\left (\frac{2}{n},- c \left (a + b x\right )^{n} \log{\left (f \right )} \right )}}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(-c*(a + b*x)**n*log(f))**(-1/n)*(a + b*x)*Gamma(1/n, -c*(a + b*x)**n*log(f))/
(b**2*n) - (-c*(a + b*x)**n*log(f))**(-2/n)*(a + b*x)**2*Gamma(2/n, -c*(a + b*x)
**n*log(f))/(b**2*n)

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Mathematica [A]  time = 0.242563, size = 91, normalized size = 0.92 \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-2/n} \left ((a+b x) \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )-a \left (-c \log (f) (a+b x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )\right )}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b*x)*((a + b*x)*Gamma[2/n, -(c*(a + b*x)^n*Log[f])] - a*Gamma[n^(-1), -(
c*(a + b*x)^n*Log[f])]*(-(c*(a + b*x)^n*Log[f]))^n^(-1)))/(b^2*n*(-(c*(a + b*x)^
n*Log[f]))^(2/n)))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{f}^{c \left ( bx+a \right ) ^{n}}x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c*(b*x+a)^n)*x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^n*c)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(f^((b*x + a)^n*c)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**(c*(a + b*x)**n)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^((b*x + a)^n*c)*x, x)