Optimal. Leaf size=41 \[ \frac {x^m f^{a c} (-b c x \log (f))^{-m} \Gamma (m+1,-b c x \log (f))}{b c \log (f)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2181} \[ \frac {x^m f^{a c} (-b c x \log (f))^{-m} \text {Gamma}(m+1,-b c x \log (f))}{b c \log (f)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rubi steps
\begin {align*} \int f^{c (a+b x)} x^m \, dx &=\frac {f^{a c} x^m \Gamma (1+m,-b c x \log (f)) (-b c x \log (f))^{-m}}{b c \log (f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 36, normalized size = 0.88 \[ x^{m+1} \left (-f^{a c}\right ) (-b c x \log (f))^{-m-1} \Gamma (m+1,-b c x \log (f)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 39, normalized size = 0.95 \[ \frac {e^{\left (a c \log \relax (f) - m \log \left (-b c \log \relax (f)\right )\right )} \Gamma \left (m + 1, -b c x \log \relax (f)\right )}{b c \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{{\left (b x + a\right )} c} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 117, normalized size = 2.85 \[ -\frac {\left (m \,x^{m} \left (-b c \right )^{m} \left (-b c x \ln \relax (f )\right )^{-m} \ln \relax (f )^{m} \Gamma \relax (m )-m \,x^{m} \left (-b c \right )^{m} \left (-b c x \ln \relax (f )\right )^{-m} \ln \relax (f )^{m} \Gamma \left (m , -b c x \ln \relax (f )\right )-x^{m} \left (-b c \right )^{m} \ln \relax (f )^{m} {\mathrm e}^{b c x \ln \relax (f )}\right ) f^{a c} \left (-b c \right )^{-m} \ln \relax (f )^{-m -1}}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 36, normalized size = 0.88 \[ -\left (-b c x \log \relax (f)\right )^{-m - 1} f^{a c} x^{m + 1} \Gamma \left (m + 1, -b c x \log \relax (f)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.50, size = 41, normalized size = 1.00 \[ \frac {f^{a\,c}\,x^m\,\Gamma \left (m+1,-b\,c\,x\,\ln \relax (f)\right )}{b\,c\,\ln \relax (f)\,{\left (-b\,c\,x\,\ln \relax (f)\right )}^m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c \left (a + b x\right )} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________