Optimal. Leaf size=68 \[ -\frac {b c \log (f) f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x} \]
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Rubi [A] time = 0.40, 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 = {2223, 6742, 2222, 2210, 2228, 2178, 2209} \[ -\frac {b c \log (f) f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2209
Rule 2210
Rule 2222
Rule 2223
Rule 2228
Rule 6742
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)) \operatorname {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 = 0.11, size = 68, normalized size = 1.00 \[ -\frac {b c \log (f) f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a^2+b x a}\right )}{a^2}-\frac {b f^{\frac {c}{a+b x}}}{a}-\frac {f^{\frac {c}{a+b x}}}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 60, normalized size = 0.88 \[ -\frac {b c f^{\frac {c}{a}} x {\rm Ei}\left (-\frac {b c x \log \relax (f)}{a b x + a^{2}}\right ) \log \relax (f) + {\left (a b x + a^{2}\right )} f^{\frac {c}{b x + a}}}{a^{2} x} \]
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 \[ \int \frac {f^{\frac {c}{b x + a}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 80, normalized size = 1.18 \[ \frac {b c \,f^{\frac {c}{a}} \Ei \left (1, -\frac {c \ln \relax (f )}{b x +a}+\frac {c \ln \relax (f )}{a}\right ) \ln \relax (f )}{a^{2}}+\frac {b c \,f^{\frac {c}{b x +a}} \ln \relax (f )}{\left (\frac {c \ln \relax (f )}{b x +a}-\frac {c \ln \relax (f )}{a}\right ) a^{2}} \]
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 \[ \int \frac {f^{\frac {c}{b x + a}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{\frac {c}{a+b\,x}}}{x^2} \,d x \]
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 \[ \int \frac {f^{\frac {c}{a + b x}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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