Optimal. Leaf size=68 \[ -\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {b c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^2} \]
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Rubi [A]
time = 0.28, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2255, 6874,
2254, 2241, 2260, 2209, 2240} \begin {gather*} -\frac {b c \log (f) f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2240
Rule 2241
Rule 2254
Rule 2255
Rule 2260
Rule 6874
Rubi steps
\begin {align*} \int \frac {f^{\frac {c}{a+b x}}}{x^2} \, dx &=-\frac {f^{\frac {c}{a+b x}}}{x}-(b c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)^2} \, dx\\ &=-\frac {f^{\frac {c}{a+b x}}}{x}-(b c \log (f)) \int \left (\frac {f^{\frac {c}{a+b x}}}{a^2 x}-\frac {b f^{\frac {c}{a+b x}}}{a (a+b x)^2}-\frac {b f^{\frac {c}{a+b x}}}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {(b c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{x} \, dx}{a^2}+\frac {\left (b^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{a^2}+\frac {\left (b^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{(a+b x)^2} \, dx}{a}\\ &=-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {b c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{a^2}-\frac {(b c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)} \, dx}{a}-\frac {\left (b^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{a^2}\\ &=-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {(b c \log (f)) \text {Subst}\left (\int \frac {f^{\frac {c}{a}-\frac {b c x}{a}}}{x} \, dx,x,\frac {x}{a+b x}\right )}{a^2}\\ &=-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {b c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 68, normalized size = 1.00 \begin {gather*} -\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x}-\frac {b c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a^2+a b x}\right ) \log (f)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 80, normalized size = 1.18
method | result | size |
risch | \(\frac {\ln \left (f \right ) b c \,f^{\frac {c}{b x +a}}}{a^{2} \left (\frac {c \ln \left (f \right )}{b x +a}-\frac {c \ln \left (f \right )}{a}\right )}+\frac {\ln \left (f \right ) b c \,f^{\frac {c}{a}} \expIntegral \left (1, -\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right )}{a^{2}}\) | \(80\) |
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 = 0.35, size = 60, normalized size = 0.88 \begin {gather*} -\frac {b c f^{\frac {c}{a}} x {\rm Ei}\left (-\frac {b c x \log \left (f\right )}{a b x + a^{2}}\right ) \log \left (f\right ) + {\left (a b x + a^{2}\right )} f^{\frac {c}{b x + a}}}{a^{2} x} \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 \frac {f^{\frac {c}{a + b x}}}{x^{2}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f^{\frac {c}{a+b\,x}}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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