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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]