Optimal. Leaf size=100 \[ \frac {(m+1) (d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{m+1}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2306, 2310, 2178} \[ \frac {(m+1) (d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{m+1}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {(1+m) \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx}{b n}\\ &=-\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left ((1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b d n^2}\\ &=\frac {e^{-\frac {a (1+m)}{b n}} (1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \text {Ei}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 89, normalized size = 0.89 \[ \frac {(d x)^m \left ((m+1) x^{-m} \exp \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n x}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 131, normalized size = 1.31 \[ -\frac {b n x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} - {\left ({\left (b m + b\right )} n \log \relax (x) + a m + {\left (b m + b\right )} \log \relax (c) + a\right )} {\rm Ei}\left (\frac {{\left (b m + b\right )} n \log \relax (x) + a m + {\left (b m + b\right )} \log \relax (c) + a}{b n}\right ) e^{\left (\frac {b m n \log \relax (d) - a m - {\left (b m + b\right )} \log \relax (c) - a}{b n}\right )}}{b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}} \]
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 \[ \int \frac {\left (d x\right )^{m}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.96, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]
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 \[ d^{m} {\left (m + 1\right )} \int \frac {x^{m}}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n}\,{d x} - \frac {d^{m} x x^{m}}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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