Optimal. Leaf size=135 \[ -\frac {6 b^3 m^3 n^3 (e x)^{1+q}}{e (1+q)^4}+\frac {6 b^2 m^2 n^2 (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^3}-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)} \]
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Rubi [A]
time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2342, 2341,
2495} \begin {gather*} \frac {6 b^2 m^2 n^2 (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (q+1)^3}+\frac {(e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (q+1)}-\frac {3 b m n (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (q+1)^2}-\frac {6 b^3 m^3 n^3 (e x)^{q+1}}{e (q+1)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
Rule 2495
Rubi steps
\begin {align*} \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3 \, dx &=\text {Subst}\left (\int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^3 \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}-\text {Subst}\left (\frac {(3 b m n) \int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^2 \, dx}{1+q},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}+\text {Subst}\left (\frac {\left (6 b^2 m^2 n^2\right ) \int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx}{(1+q)^2},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {6 b^3 m^3 n^3 (e x)^{1+q}}{e (1+q)^4}+\frac {6 b^2 m^2 n^2 (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^3}-\frac {3 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3}{e (1+q)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 91, normalized size = 0.67 \begin {gather*} \frac {x (e x)^q \left (\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^3-\frac {3 b m n \left ((1+q)^2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2+2 b m n \left (b m n-(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )\right )\right )}{(1+q)^3}\right )}{1+q} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{q} \left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs.
\(2 (135) = 270\).
time = 0.29, size = 285, normalized size = 2.11 \begin {gather*} \frac {\left (x e\right )^{q + 1} b^{3} e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{3}}{q + 1} + \frac {3 \, \left (x e\right )^{q + 1} a b^{2} e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{q + 1} - \frac {3 \, a^{2} b m n x e^{\left (q \log \left (x\right ) + q\right )}}{{\left (q + 1\right )}^{2}} + \frac {3 \, \left (x e\right )^{q + 1} a^{2} b e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{q + 1} + 6 \, {\left (\frac {m^{2} n^{2} x e^{\left (q \log \left (x\right ) + q\right )}}{{\left (q + 1\right )}^{3}} - \frac {m n x e^{\left (q \log \left (x\right ) + q\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} a b^{2} - 3 \, {\left (\frac {m n x e^{\left (q \log \left (x\right ) + q\right )} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{{\left (q + 1\right )}^{2}} + \frac {2 \, {\left (\frac {m^{2} n^{2} x e^{\left (q \log \left (x\right ) + q + 1\right )}}{{\left (q + 1\right )}^{3}} - \frac {m n x e^{\left (q \log \left (x\right ) + q + 1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} m n e^{\left (-1\right )}}{q + 1}\right )} b^{3} + \frac {\left (x e\right )^{q + 1} a^{3} e^{\left (-1\right )}}{q + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1355 vs.
\(2 (135) = 270\).
time = 0.35, size = 1355, normalized size = 10.04 \begin {gather*} \frac {{\left ({\left (b^{3} q^{3} + 3 \, b^{3} q^{2} + 3 \, b^{3} q + b^{3}\right )} x \log \left (c\right )^{3} + {\left (b^{3} n^{3} q^{3} + 3 \, b^{3} n^{3} q^{2} + 3 \, b^{3} n^{3} q + b^{3} n^{3}\right )} x \log \left (d\right )^{3} + {\left (b^{3} m^{3} n^{3} q^{3} + 3 \, b^{3} m^{3} n^{3} q^{2} + 3 \, b^{3} m^{3} n^{3} q + b^{3} m^{3} n^{3}\right )} x \log \left (x\right )^{3} + 3 \, {\left (a b^{2} q^{3} - b^{3} m n + a b^{2} - {\left (b^{3} m n - 3 \, a b^{2}\right )} q^{2} - {\left (2 \, b^{3} m n - 3 \, a b^{2}\right )} q\right )} x \log \left (c\right )^{2} + 3 \, {\left (2 \, b^{3} m^{2} n^{2} + a^{2} b q^{3} - 2 \, a b^{2} m n + a^{2} b - {\left (2 \, a b^{2} m n - 3 \, a^{2} b\right )} q^{2} + {\left (2 \, b^{3} m^{2} n^{2} - 4 \, a b^{2} m n + 3 \, a^{2} b\right )} q\right )} x \log \left (c\right ) + 3 \, {\left ({\left (b^{3} n^{2} q^{3} + 3 \, b^{3} n^{2} q^{2} + 3 \, b^{3} n^{2} q + b^{3} n^{2}\right )} x \log \left (c\right ) + {\left (a b^{2} n^{2} q^{3} - b^{3} m n^{3} + a b^{2} n^{2} - {\left (b^{3} m n^{3} - 3 \, a b^{2} n^{2}\right )} q^{2} - {\left (2 \, b^{3} m n^{3} - 3 \, a b^{2} n^{2}\right )} q\right )} x\right )} \log \left (d\right )^{2} + 3 \, {\left ({\left (b^{3} m^{2} n^{2} q^{3} + 3 \, b^{3} m^{2} n^{2} q^{2} + 3 \, b^{3} m^{2} n^{2} q + b^{3} m^{2} n^{2}\right )} x \log \left (c\right ) + {\left (b^{3} m^{2} n^{3} q^{3} + 3 \, b^{3} m^{2} n^{3} q^{2} + 3 \, b^{3} m^{2} n^{3} q + b^{3} m^{2} n^{3}\right )} x \log \left (d\right ) + {\left (a b^{2} m^{2} n^{2} q^{3} - b^{3} m^{3} n^{3} + a b^{2} m^{2} n^{2} - {\left (b^{3} m^{3} n^{3} - 3 \, a b^{2} m^{2} n^{2}\right )} q^{2} - {\left (2 \, b^{3} m^{3} n^{3} - 3 \, a b^{2} m^{2} n^{2}\right )} q\right )} x\right )} \log \left (x\right )^{2} - {\left (6 \, b^{3} m^{3} n^{3} - 6 \, a b^{2} m^{2} n^{2} - a^{3} q^{3} + 3 \, a^{2} b m n - a^{3} + 3 \, {\left (a^{2} b m n - a^{3}\right )} q^{2} - 3 \, {\left (2 \, a b^{2} m^{2} n^{2} - 2 \, a^{2} b m n + a^{3}\right )} q\right )} x + 3 \, {\left ({\left (b^{3} n q^{3} + 3 \, b^{3} n q^{2} + 3 \, b^{3} n q + b^{3} n\right )} x \log \left (c\right )^{2} + 2 \, {\left (a b^{2} n q^{3} - b^{3} m n^{2} + a b^{2} n - {\left (b^{3} m n^{2} - 3 \, a b^{2} n\right )} q^{2} - {\left (2 \, b^{3} m n^{2} - 3 \, a b^{2} n\right )} q\right )} x \log \left (c\right ) + {\left (2 \, b^{3} m^{2} n^{3} + a^{2} b n q^{3} - 2 \, a b^{2} m n^{2} + a^{2} b n - {\left (2 \, a b^{2} m n^{2} - 3 \, a^{2} b n\right )} q^{2} + {\left (2 \, b^{3} m^{2} n^{3} - 4 \, a b^{2} m n^{2} + 3 \, a^{2} b n\right )} q\right )} x\right )} \log \left (d\right ) + 3 \, {\left ({\left (b^{3} m n q^{3} + 3 \, b^{3} m n q^{2} + 3 \, b^{3} m n q + b^{3} m n\right )} x \log \left (c\right )^{2} + {\left (b^{3} m n^{3} q^{3} + 3 \, b^{3} m n^{3} q^{2} + 3 \, b^{3} m n^{3} q + b^{3} m n^{3}\right )} x \log \left (d\right )^{2} + 2 \, {\left (a b^{2} m n q^{3} - b^{3} m^{2} n^{2} + a b^{2} m n - {\left (b^{3} m^{2} n^{2} - 3 \, a b^{2} m n\right )} q^{2} - {\left (2 \, b^{3} m^{2} n^{2} - 3 \, a b^{2} m n\right )} q\right )} x \log \left (c\right ) + {\left (2 \, b^{3} m^{3} n^{3} + a^{2} b m n q^{3} - 2 \, a b^{2} m^{2} n^{2} + a^{2} b m n - {\left (2 \, a b^{2} m^{2} n^{2} - 3 \, a^{2} b m n\right )} q^{2} + {\left (2 \, b^{3} m^{3} n^{3} - 4 \, a b^{2} m^{2} n^{2} + 3 \, a^{2} b m n\right )} q\right )} x + 2 \, {\left ({\left (b^{3} m n^{2} q^{3} + 3 \, b^{3} m n^{2} q^{2} + 3 \, b^{3} m n^{2} q + b^{3} m n^{2}\right )} x \log \left (c\right ) + {\left (a b^{2} m n^{2} q^{3} - b^{3} m^{2} n^{3} + a b^{2} m n^{2} - {\left (b^{3} m^{2} n^{3} - 3 \, a b^{2} m n^{2}\right )} q^{2} - {\left (2 \, b^{3} m^{2} n^{3} - 3 \, a b^{2} m n^{2}\right )} q\right )} x\right )} \log \left (d\right )\right )} \log \left (x\right )\right )} e^{\left (q \log \left (x\right ) + q\right )}}{q^{4} + 4 \, q^{3} + 6 \, q^{2} + 4 \, q + 1} \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 \left (e x\right )^{q} \left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{3}\, 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. 1811 vs.
\(2 (135) = 270\).
time = 4.39, size = 1811, normalized size = 13.41 \begin {gather*} \text {Too large to display} \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 {\left (e\,x\right )}^q\,{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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