Optimal. Leaf size=33 \[ \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)} \]
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Rubi [A]
time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2339, 30, 2495}
\begin {gather*} \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2339
Rule 2495
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \log \left (c d^n x^{m n}\right )\right )^p}{x} \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\text {Subst}\left (\frac {\text {Subst}\left (\int x^p \, dx,x,a+b \log \left (c d^n x^{m n}\right )\right )}{b m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 34, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{1+p}}{b m n \left (1+p \right )}\) | \(34\) |
default | \(\frac {\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{1+p}}{b m n \left (1+p \right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 1.00 \begin {gather*} \frac {{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p + 1}}{b m n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 49, normalized size = 1.48 \begin {gather*} \frac {{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )} {\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p}}{b m n p + b m n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.46, size = 80, normalized size = 2.42 \begin {gather*} - \begin {cases} - a^{p} \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (a + b \log {\left (c d^{n} \right )}\right )^{p} \log {\left (x \right )} & \text {for}\: m = 0 \\- \left (a + b \log {\left (c \right )}\right )^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )} \right )} & \text {otherwise} \end {cases}}{b m n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.62, size = 36, normalized size = 1.09 \begin {gather*} \frac {{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p + 1}}{b m n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.07, size = 33, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^{p+1}}{b\,m\,n\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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