Optimal. Leaf size=41 \[ f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2222, 2210, 2228, 2178} \[ f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2210
Rule 2222
Rule 2228
Rubi steps
\begin {align*} \int \frac {f^{\frac {c}{a+b x}}}{x} \, dx &=a \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)} \, dx+b \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx\\ &=-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right )+\operatorname {Subst}\left (\int \frac {f^{\frac {c}{a}-\frac {b c x}{a}}}{x} \, dx,x,\frac {x}{a+b x}\right )\\ &=-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right )+f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \[ f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a^2+b x a}\right )-\text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 41, normalized size = 1.00 \[ f^{\frac {c}{a}} {\rm Ei}\left (-\frac {b c x \log \relax (f)}{a b x + a^{2}}\right ) - {\rm Ei}\left (\frac {c \log \relax (f)}{b x + a}\right ) \]
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 \[ \int \frac {f^{\frac {c}{b x + a}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 47, normalized size = 1.15 \[ -f^{\frac {c}{a}} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}+\frac {c \ln \relax (f )}{a}\right )+\Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}\right ) \]
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 \[ \int \frac {f^{\frac {c}{b x + a}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {f^{\frac {c}{a+b\,x}}}{x} \,d x \]
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 \[ \int \frac {f^{\frac {c}{a + b x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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