Optimal. Leaf size=81 \[ \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac {2 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac {2 b^2 n^2 (d x)^{m+1}}{d (m+1)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2305, 2304} \[ \frac {(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)}-\frac {2 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^2}+\frac {2 b^2 n^2 (d x)^{m+1}}{d (m+1)^3} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)}-\frac {(2 b n) \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx}{1+m}\\ &=\frac {2 b^2 n^2 (d x)^{1+m}}{d (1+m)^3}-\frac {2 b n (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right )^2}{d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 0.94 \[ \frac {x (d x)^m \left (a^2 (m+1)^2+2 b (m+1) (a m+a-b n) \log \left (c x^n\right )-2 a b (m+1) n+b^2 (m+1)^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{(m+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 208, normalized size = 2.57 \[ \frac {{\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} x \log \relax (x)^{2} + {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} x \log \relax (c)^{2} + 2 \, {\left (a b m^{2} + 2 \, a b m + a b - {\left (b^{2} m + b^{2}\right )} n\right )} x \log \relax (c) + {\left (a^{2} m^{2} + 2 \, b^{2} n^{2} + 2 \, a^{2} m + a^{2} - 2 \, {\left (a b m + a b\right )} n\right )} x + 2 \, {\left ({\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n x \log \relax (c) - {\left ({\left (b^{2} m + b^{2}\right )} n^{2} - {\left (a b m^{2} + 2 \, a b m + a b\right )} n\right )} x\right )} \log \relax (x)\right )} e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 402, normalized size = 4.96 \[ \frac {b^{2} d^{m} m^{2} n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} m n^{2} x x^{m} \log \relax (x)}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} m n x x^{m} \log \relax (c) \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {b^{2} d^{m} n^{2} x x^{m} \log \relax (x)^{2}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, a b d^{m} m n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {2 \, b^{2} d^{m} n^{2} x x^{m} \log \relax (x)}{m^{3} + 3 \, m^{2} + 3 \, m + 1} + \frac {2 \, b^{2} d^{m} n x x^{m} \log \relax (c) \log \relax (x)}{m^{2} + 2 \, m + 1} + \frac {2 \, b^{2} d^{m} n^{2} x x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \frac {2 \, b^{2} d^{m} n x x^{m} \log \relax (c)}{m^{2} + 2 \, m + 1} + \frac {2 \, a b d^{m} n x x^{m} \log \relax (x)}{m^{2} + 2 \, m + 1} - \frac {2 \, a b d^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (d x\right )^{m} b^{2} x \log \relax (c)^{2}}{m + 1} + \frac {2 \, \left (d x\right )^{m} a b x \log \relax (c)}{m + 1} + \frac {\left (d x\right )^{m} a^{2} x}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 2126, normalized size = 26.25 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 132, normalized size = 1.63 \[ -\frac {2 \, a b d^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} - 2 \, {\left (\frac {d^{m} n x x^{m} \log \left (c x^{n}\right )}{{\left (m + 1\right )}^{2}} - \frac {d^{m} n^{2} x x^{m}}{{\left (m + 1\right )}^{3}}\right )} b^{2} + \frac {\left (d x\right )^{m + 1} b^{2} \log \left (c x^{n}\right )^{2}}{d {\left (m + 1\right )}} + \frac {2 \, \left (d x\right )^{m + 1} a b \log \left (c x^{n}\right )}{d {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.73, size = 891, normalized size = 11.00 \[ \begin {cases} \frac {a^{2} d^{m} m^{2} x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a^{2} d^{m} m x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {a^{2} d^{m} x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} m^{2} n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} m^{2} x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 a b d^{m} m n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b d^{m} m n x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 a b d^{m} m x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} n x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 a b d^{m} n x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 a b d^{m} x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} m^{2} n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m^{2} n x x^{m} \log {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} m^{2} x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} m n^{2} x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {4 b^{2} d^{m} m n x x^{m} \log {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} m n x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} m x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} n^{2} x x^{m} \log {\relax (x )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} n^{2} x x^{m} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} n^{2} x x^{m}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} d^{m} n x x^{m} \log {\relax (c )} \log {\relax (x )}}{m^{3} + 3 m^{2} + 3 m + 1} - \frac {2 b^{2} d^{m} n x x^{m} \log {\relax (c )}}{m^{3} + 3 m^{2} + 3 m + 1} + \frac {b^{2} d^{m} x x^{m} \log {\relax (c )}^{2}}{m^{3} + 3 m^{2} + 3 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\relax (c )} + b^{2} \log {\relax (c )}^{2}\right ) \log {\relax (x )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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