Optimal. Leaf size=41 \[ \frac {f^{\frac {c}{a+b x}} (a+b x)}{b}-\frac {c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2237, 2241}
\begin {gather*} \frac {(a+b x) f^{\frac {c}{a+b x}}}{b}-\frac {c \log (f) \text {Ei}\left (\frac {c \log (f)}{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2237
Rule 2241
Rubi steps
\begin {align*} \int f^{\frac {c}{a+b x}} \, dx &=\frac {f^{\frac {c}{a+b x}} (a+b x)}{b}+(c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx\\ &=\frac {f^{\frac {c}{a+b x}} (a+b x)}{b}-\frac {c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} \frac {f^{\frac {c}{a+b x}} (a+b x)}{b}-\frac {c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 52, normalized size = 1.27
method | result | size |
risch | \(f^{\frac {c}{b x +a}} x +\frac {f^{\frac {c}{b x +a}} a}{b}+\frac {c \ln \left (f \right ) \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}\right )}{b}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 40, normalized size = 0.98 \begin {gather*} -\frac {c {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right ) \log \left (f\right ) - {\left (b x + a\right )} f^{\frac {c}{b x + a}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{\frac {c}{a + b x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.55, size = 50, normalized size = 1.22 \begin {gather*} f^{\frac {c}{a+b\,x}}\,x+\frac {a\,f^{\frac {c}{a+b\,x}}}{b}-\frac {c\,\mathrm {ei}\left (\frac {c\,\ln \left (f\right )}{a+b\,x}\right )\,\ln \left (f\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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