Optimal. Leaf size=127 \[ \frac {(q+1) (e x)^{q+1} e^{-\frac {a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac {q+1}{m n}} \text {Ei}\left (\frac {(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac {(e x)^{q+1}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2306, 2310, 2178, 2445} \[ \frac {(q+1) (e x)^{q+1} e^{-\frac {a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac {q+1}{m n}} \text {Ei}\left (\frac {(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac {(e x)^{q+1}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rule 2445
Rubi steps
\begin {align*} \int \frac {(e x)^q}{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {(e x)^q}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^2} \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {(e x)^{1+q}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}+\operatorname {Subst}\left (\frac {(1+q) \int \frac {(e x)^q}{a+b \log \left (c d^n x^{m n}\right )} \, dx}{b m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {(e x)^{1+q}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}+\operatorname {Subst}\left (\frac {\left ((1+q) (e x)^{1+q} \left (c d^n x^{m n}\right )^{-\frac {1+q}{m n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+q) x}{m n}}}{a+b x} \, dx,x,\log \left (c d^n x^{m n}\right )\right )}{b e m^2 n^2},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac {e^{-\frac {a (1+q)}{b m n}} (1+q) (e x)^{1+q} \left (c \left (d x^m\right )^n\right )^{-\frac {1+q}{m n}} \text {Ei}\left (\frac {(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac {(e x)^{1+q}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 112, normalized size = 0.88 \[ \frac {(e x)^q \left ((q+1) x^{-q} \exp \left (-\frac {(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )-b m n \log (x)\right )}{b m n}\right ) \text {Ei}\left (\frac {(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )-\frac {b m n x}{a+b \log \left (c \left (d x^m\right )^n\right )}\right )}{b^2 m^2 n^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 202, normalized size = 1.59 \[ -\frac {b m n x e^{\left (q \log \relax (e) + q \log \relax (x)\right )} - {\left (a q + {\left (b q + b\right )} \log \relax (c) + {\left (b n q + b n\right )} \log \relax (d) + {\left (b m n q + b m n\right )} \log \relax (x) + a\right )} {\rm Ei}\left (\frac {a q + {\left (b q + b\right )} \log \relax (c) + {\left (b n q + b n\right )} \log \relax (d) + {\left (b m n q + b m n\right )} \log \relax (x) + a}{b m n}\right ) e^{\left (\frac {b m n q \log \relax (e) - a q - {\left (b q + b\right )} \log \relax (c) - {\left (b n q + b n\right )} \log \relax (d) - a}{b m n}\right )}}{b^{3} m^{3} n^{3} \log \relax (x) + b^{3} m^{2} n^{3} \log \relax (d) + b^{3} m^{2} n^{2} \log \relax (c) + a b^{2} m^{2} n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.84, size = 1540, normalized size = 12.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{q}}{\left (b \ln \left (c \left (d \,x^{m}\right )^{n}\right )+a \right )^{2}}\, dx \]
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 \[ e^{q} {\left (q + 1\right )} \int \frac {x^{q}}{b^{2} m n \log \left ({\left (x^{m}\right )}^{n}\right ) + a b m n + {\left (m n^{2} \log \relax (d) + m n \log \relax (c)\right )} b^{2}}\,{d x} - \frac {e^{q} x x^{q}}{b^{2} m n \log \left ({\left (x^{m}\right )}^{n}\right ) + a b m n + {\left (m n^{2} \log \relax (d) + m n \log \relax (c)\right )} b^{2}} \]
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 \[ \int \frac {{\left (e\,x\right )}^q}{{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (e x\right )^{q}}{\left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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