Optimal. Leaf size=93 \[ \frac {(e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (q+1)}-\frac {2 b m n (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (q+1)^2}+\frac {2 b^2 m^2 n^2 (e x)^{q+1}}{e (q+1)^3} \]
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Rubi [A] time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2305, 2304, 2445} \[ \frac {(e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (q+1)}-\frac {2 b m n (e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (q+1)^2}+\frac {2 b^2 m^2 n^2 (e x)^{q+1}}{e (q+1)^3} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rule 2445
Rubi steps
\begin {align*} \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2 \, dx &=\operatorname {Subst}\left (\int (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} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)}-\operatorname {Subst}\left (\frac {(2 b m n) \int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx}{1+q},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac {2 b^2 m^2 n^2 (e x)^{1+q}}{e (1+q)^3}-\frac {2 b m n (e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{e (1+q)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 90, normalized size = 0.97 \[ \frac {x (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^2}{q+1}-\frac {2 b m n x^{-q} (e x)^q \left (\frac {x^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{q+1}-\frac {b m n x^{q+1}}{(q+1)^2}\right )}{q+1} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 391, normalized size = 4.20 \[ \frac {{\left ({\left (b^{2} q^{2} + 2 \, b^{2} q + b^{2}\right )} x \log \relax (c)^{2} + {\left (b^{2} n^{2} q^{2} + 2 \, b^{2} n^{2} q + b^{2} n^{2}\right )} x \log \relax (d)^{2} + {\left (b^{2} m^{2} n^{2} q^{2} + 2 \, b^{2} m^{2} n^{2} q + b^{2} m^{2} n^{2}\right )} x \log \relax (x)^{2} - 2 \, {\left (b^{2} m n - a b q^{2} - a b + {\left (b^{2} m n - 2 \, a b\right )} q\right )} x \log \relax (c) + {\left (2 \, b^{2} m^{2} n^{2} - 2 \, a b m n + a^{2} q^{2} + a^{2} - 2 \, {\left (a b m n - a^{2}\right )} q\right )} x + 2 \, {\left ({\left (b^{2} n q^{2} + 2 \, b^{2} n q + b^{2} n\right )} x \log \relax (c) - {\left (b^{2} m n^{2} - a b n q^{2} - a b n + {\left (b^{2} m n^{2} - 2 \, a b n\right )} q\right )} x\right )} \log \relax (d) + 2 \, {\left ({\left (b^{2} m n q^{2} + 2 \, b^{2} m n q + b^{2} m n\right )} x \log \relax (c) + {\left (b^{2} m n^{2} q^{2} + 2 \, b^{2} m n^{2} q + b^{2} m n^{2}\right )} x \log \relax (d) - {\left (b^{2} m^{2} n^{2} - a b m n q^{2} - a b m n + {\left (b^{2} m^{2} n^{2} - 2 \, a b m n\right )} q\right )} x\right )} \log \relax (x)\right )} e^{\left (q \log \relax (e) + q \log \relax (x)\right )}}{q^{3} + 3 \, q^{2} + 3 \, q + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 561, normalized size = 6.03 \[ \frac {b^{2} m^{2} n^{2} q^{2} x x^{q} e^{q} \log \relax (x)^{2}}{q^{3} + 3 \, q^{2} + 3 \, q + 1} + \frac {2 \, b^{2} m^{2} n^{2} q x x^{q} e^{q} \log \relax (x)^{2}}{q^{3} + 3 \, q^{2} + 3 \, q + 1} - \frac {2 \, b^{2} m^{2} n^{2} q x x^{q} e^{q} \log \relax (x)}{q^{3} + 3 \, q^{2} + 3 \, q + 1} + \frac {2 \, b^{2} m n^{2} q x x^{q} e^{q} \log \relax (d) \log \relax (x)}{q^{2} + 2 \, q + 1} + \frac {b^{2} m^{2} n^{2} x x^{q} e^{q} \log \relax (x)^{2}}{q^{3} + 3 \, q^{2} + 3 \, q + 1} - \frac {2 \, b^{2} m^{2} n^{2} x x^{q} e^{q} \log \relax (x)}{q^{3} + 3 \, q^{2} + 3 \, q + 1} + \frac {2 \, b^{2} m n q x x^{q} e^{q} \log \relax (c) \log \relax (x)}{q^{2} + 2 \, q + 1} + \frac {2 \, b^{2} m n^{2} x x^{q} e^{q} \log \relax (d) \log \relax (x)}{q^{2} + 2 \, q + 1} + \frac {2 \, b^{2} m^{2} n^{2} x x^{q} e^{q}}{q^{3} + 3 \, q^{2} + 3 \, q + 1} - \frac {2 \, b^{2} m n^{2} x x^{q} e^{q} \log \relax (d)}{q^{2} + 2 \, q + 1} + \frac {b^{2} n^{2} x x^{q} e^{q} \log \relax (d)^{2}}{q + 1} + \frac {2 \, a b m n q x x^{q} e^{q} \log \relax (x)}{q^{2} + 2 \, q + 1} + \frac {2 \, b^{2} m n x x^{q} e^{q} \log \relax (c) \log \relax (x)}{q^{2} + 2 \, q + 1} - \frac {2 \, b^{2} m n x x^{q} e^{q} \log \relax (c)}{q^{2} + 2 \, q + 1} + \frac {2 \, b^{2} n x x^{q} e^{q} \log \relax (c) \log \relax (d)}{q + 1} + \frac {2 \, a b m n x x^{q} e^{q} \log \relax (x)}{q^{2} + 2 \, q + 1} - \frac {2 \, a b m n x x^{q} e^{q}}{q^{2} + 2 \, q + 1} + \frac {b^{2} x x^{q} e^{q} \log \relax (c)^{2}}{q + 1} + \frac {2 \, a b n x x^{q} e^{q} \log \relax (d)}{q + 1} + \frac {2 \, a b x x^{q} e^{q} \log \relax (c)}{q + 1} + \frac {a^{2} x x^{q} e^{q}}{q + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d \,x^{m}\right )^{n}\right )+a \right )^{2} \left (e x \right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 149, normalized size = 1.60 \[ -\frac {2 \, a b e^{q} m n x x^{q}}{{\left (q + 1\right )}^{2}} + 2 \, {\left (\frac {e^{q} m^{2} n^{2} x x^{q}}{{\left (q + 1\right )}^{3}} - \frac {e^{q} m n x x^{q} \log \left (\left (d x^{m}\right )^{n} c\right )}{{\left (q + 1\right )}^{2}}\right )} b^{2} + \frac {\left (e x\right )^{q + 1} b^{2} \log \left (\left (d x^{m}\right )^{n} c\right )^{2}}{e {\left (q + 1\right )}} + \frac {2 \, \left (e x\right )^{q + 1} a b \log \left (\left (d x^{m}\right )^{n} c\right )}{e {\left (q + 1\right )}} + \frac {\left (e x\right )^{q + 1} a^{2}}{e {\left (q + 1\right )}} \]
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 \[ \int {\left (e\,x\right )}^q\,{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^2 \,d x \]
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 \[ \int \left (e x\right )^{q} \left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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