Optimal. Leaf size=127 \[ \frac {e^{-\frac {a (1+q)}{b m n}} (1+q) (e x)^{1+q} \left (c \left (d x^m\right )^n\right )^{-\frac {1+q}{m n}} \text {Ei}\left (\frac {(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{b^2 e m^2 n^2}-\frac {(e x)^{1+q}}{b e m n \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )} \]
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Rubi [A]
time = 0.17, 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 = {2343, 2347,
2209, 2495} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2347
Rule 2495
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 &=\text {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 )}+\text {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 )}+\text {Subst}\left (\frac {\left ((1+q) (e x)^{1+q} \left (c d^n x^{m n}\right )^{-\frac {1+q}{m n}}\right ) \text {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 = 0.19, size = 112, normalized size = 0.88 \begin {gather*} \frac {(e x)^q \left (e^{-\frac {(1+q) \left (a-b m n \log (x)+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}} (1+q) x^{-q} \text {Ei}\left (\frac {(1+q) \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{q}}{\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 198, normalized size = 1.56 \begin {gather*} -\frac {b m n x e^{\left (q \log \left (x\right ) + q\right )} - {\left (a q + {\left (b q + b\right )} \log \left (c\right ) + {\left (b n q + b n\right )} \log \left (d\right ) + {\left (b m n q + b m n\right )} \log \left (x\right ) + a\right )} {\rm Ei}\left (\frac {a q + {\left (b q + b\right )} \log \left (c\right ) + {\left (b n q + b n\right )} \log \left (d\right ) + {\left (b m n q + b m n\right )} \log \left (x\right ) + a}{b m n}\right ) e^{\left (\frac {{\left (b m n - a\right )} q - {\left (b q + b\right )} \log \left (c\right ) - {\left (b n q + b n\right )} \log \left (d\right ) - a}{b m n}\right )}}{b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\left (e x\right )^{q}}{\left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1540 vs.
\(2 (127) = 254\).
time = 4.49, size = 1540, normalized size = 12.13 \begin {gather*} -\frac {b m n x x^{q} e^{q}}{b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}} + \frac {b m n q {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (x\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {b n q {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (d\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {b m n {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (x\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {b q {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (c\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {b n {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (d\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {a q {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )}}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {b {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )} \log \left (c\right )}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} + \frac {a {\rm Ei}\left (q \log \left (x\right ) + \frac {q \log \left (d\right )}{m} + \frac {q \log \left (c\right )}{m n} + \frac {\log \left (d\right )}{m} + \frac {a q}{b m n} + \frac {\log \left (c\right )}{m n} + \frac {a}{b m n} + \log \left (x\right )\right ) e^{\left (q - \frac {a q}{b m n} - \frac {a}{b m n}\right )}}{{\left (b^{3} m^{3} n^{3} \log \left (x\right ) + b^{3} m^{2} n^{3} \log \left (d\right ) + b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} c^{\frac {q}{m n}} c^{\frac {1}{m n}} d^{\frac {q}{m}} d^{\left (\frac {1}{m}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e\,x\right )}^q}{{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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