3.242 \(\int f^{c (a+b x)} x^m \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

-f^(a*c)*(-c*b)^(-m)*ln(f)^(-m-1)/c/b*(x^m*(-c*b)^m*ln(f)^m*m*GAMMA(m)*(-b*c*x*l
n(f))^(-m)-x^m*(-c*b)^m*ln(f)^m*exp(b*c*x*ln(f))-x^m*(-c*b)^m*ln(f)^m*m*(-b*c*x*
ln(f))^(-m)*GAMMA(m,-b*c*x*ln(f)))

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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]

-(-b*c*x*log(f))^(-m - 1)*f^(a*c)*x^(m + 1)*gamma(m + 1, -b*c*x*log(f))

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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]

e^(a*c*log(f) - m*log(-b*c*log(f)))*gamma(m + 1, -b*c*x*log(f))/(b*c*log(f))

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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]

Integral(f**(c*(a + b*x))*x**m, x)

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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]

integrate(f^((b*x + a)*c)*x^m, x)