Optimal. Leaf size=142 \[ \frac {(m+1)^2 (d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 d n^3}-\frac {(m+1) (d x)^{m+1}}{2 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {(d x)^{m+1}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.14, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2306, 2310, 2178} \[ \frac {(m+1)^2 (d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b^3 d n^3}-\frac {(m+1) (d x)^{m+1}}{2 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {(d x)^{m+1}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2} \]
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 )^3} \, dx &=-\frac {(d x)^{1+m}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {(1+m) \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac {(d x)^{1+m}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {(1+m) (d x)^{1+m}}{2 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {(1+m)^2 \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx}{2 b^2 n^2}\\ &=-\frac {(d x)^{1+m}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {(1+m) (d x)^{1+m}}{2 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}+\frac {\left ((1+m)^2 (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 )}{2 b^2 d n^3}\\ &=\frac {e^{-\frac {a (1+m)}{b n}} (1+m)^2 (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 )}{2 b^3 d n^3}-\frac {(d x)^{1+m}}{2 b d n \left (a+b \log \left (c x^n\right )\right )^2}-\frac {(1+m) (d x)^{1+m}}{2 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 113, normalized size = 0.80 \[ \frac {(d x)^m \left ((m+1)^2 x^{-m} \exp \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n x \left (a m+a+b (m+1) \log \left (c x^n\right )+b n\right )}{\left (a+b \log \left (c x^n\right )\right )^2}\right )}{2 b^3 n^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 322, normalized size = 2.27 \[ \frac {{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} \log \relax (x)^{2} + a^{2} m^{2} + 2 \, a^{2} m + {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} \log \relax (c)^{2} + a^{2} + 2 \, {\left (a b m^{2} + 2 \, a b m + a b\right )} \log \relax (c) + 2 \, {\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n \log \relax (c) + {\left (a b m^{2} + 2 \, a b m + a b\right )} n\right )} \log \relax (x)\right )} {\rm Ei}\left (\frac {{\left (b m + b\right )} n \log \relax (x) + a m + {\left (b m + b\right )} \log \relax (c) + a}{b n}\right ) e^{\left (\frac {b m n \log \relax (d) - a m - {\left (b m + b\right )} \log \relax (c) - a}{b n}\right )} - {\left ({\left (b^{2} m + b^{2}\right )} n^{2} x \log \relax (x) + {\left (b^{2} m + b^{2}\right )} n x \log \relax (c) + {\left (b^{2} n^{2} + {\left (a b m + a b\right )} n\right )} x\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{2 \, {\left (b^{5} n^{5} \log \relax (x)^{2} + b^{5} n^{3} \log \relax (c)^{2} + 2 \, a b^{4} n^{3} \log \relax (c) + a^{2} b^{3} n^{3} + 2 \, {\left (b^{5} n^{4} \log \relax (c) + a b^{4} n^{4}\right )} \log \relax (x)\right )}} \]
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 \[ \int \frac {\left (d x\right )^{m}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.84, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}\, dx \]
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 \[ {\left (m^{2} + 2 \, m + 1\right )} d^{m} \int \frac {x^{m}}{2 \, {\left (b^{3} n^{2} \log \relax (c) + b^{3} n^{2} \log \left (x^{n}\right ) + a b^{2} n^{2}\right )}}\,{d x} - \frac {b d^{m} {\left (m + 1\right )} x x^{m} \log \left (x^{n}\right ) + {\left (a d^{m} {\left (m + 1\right )} + {\left (d^{m} {\left (m + 1\right )} \log \relax (c) + d^{m} n\right )} b\right )} x x^{m}}{2 \, {\left (b^{4} n^{2} \log \relax (c)^{2} + b^{4} n^{2} \log \left (x^{n}\right )^{2} + 2 \, a b^{3} n^{2} \log \relax (c) + a^{2} b^{2} n^{2} + 2 \, {\left (b^{4} n^{2} \log \relax (c) + a b^{3} n^{2}\right )} \log \left (x^{n}\right )\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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