Optimal. Leaf size=41 \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]
[Out]
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Rubi [A] time = 0.041154, antiderivative size = 41, 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 \[ \frac{x^m f^{a c} (-b c x \log (f))^{-m} \text{Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]
Antiderivative was successfully verified.
[In] Int[f^(c*(a + b*x))*x^m,x]
[Out]
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Rubi in Sympy [A] time = 4.00945, size = 39, normalized size = 0.95 \[ \frac{f^{a c} x^{m} \left (- b c x \log{\left (f \right )}\right )^{- m} \Gamma{\left (m + 1,- b c x \log{\left (f \right )} \right )}}{b c \log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(c*(b*x+a))*x**m,x)
[Out]
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Mathematica [A] time = 0.0198949, size = 36, normalized size = 0.88 \[ x^{m+1} \left (-f^{a c}\right ) (-b c x \log (f))^{-m-1} \text{Gamma}(m+1,-b c x \log (f)) \]
Antiderivative was successfully verified.
[In] Integrate[f^(c*(a + b*x))*x^m,x]
[Out]
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Maple [B] time = 0.049, size = 117, normalized size = 2.9 \[ -{\frac{{f}^{ac} \left ( -cb \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m-1} \left ({x}^{m} \left ( -cb \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m\Gamma \left ( m \right ) \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}-{x}^{m} \left ( -cb \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}{{\rm e}^{bcx\ln \left ( f \right ) }}-{x}^{m} \left ( -cb \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{m}m \left ( -bcx\ln \left ( f \right ) \right ) ^{-m}\Gamma \left ( m,-bcx\ln \left ( f \right ) \right ) \right ) }{cb}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(c*(b*x+a))*x^m,x)
[Out]
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Maxima [A] time = 0.864797, size = 49, normalized size = 1.2 \[ -\left (-b c x \log \left (f\right )\right )^{-m - 1} f^{a c} x^{m + 1} \Gamma \left (m + 1, -b c x \log \left (f\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^((b*x + a)*c)*x^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265824, size = 53, normalized size = 1.29 \[ \frac{e^{\left (a c \log \left (f\right ) - m \log \left (-b c \log \left (f\right )\right )\right )} \Gamma \left (m + 1, -b c x \log \left (f\right )\right )}{b c \log \left (f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^((b*x + a)*c)*x^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int f^{c \left (a + b x\right )} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(c*(b*x+a))*x**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{{\left (b x + a\right )} c} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^((b*x + a)*c)*x^m,x, algorithm="giac")
[Out]