Optimal. Leaf size=78 \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {a+b \log \left (c x^n\right )}{b n}\right )}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2310, 2181} \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,\frac {a+b \log \left (c x^n\right )}{b n}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2310
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int e^{-\frac {x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n x}\\ &=-\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \Gamma \left (1+p,\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 78, normalized size = 1.00 \[ -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {a+b \log \left (c x^n\right )}{b n}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{p}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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