Optimal. Leaf size=19 \[ \frac {\log \left (a+b x^{1+m}\right )}{b (1+m)} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266}
\begin {gather*} \frac {\log \left (a+b x^{m+1}\right )}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rubi steps
\begin {align*} \int \frac {x^m}{a+b x^{1+m}} \, dx &=\frac {\log \left (a+b x^{1+m}\right )}{b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+b x^{1+m}\right )}{b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 21, normalized size = 1.11
method | result | size |
norman | \(\frac {\ln \left (a +b x \,{\mathrm e}^{m \ln \left (x \right )}\right )}{b \left (1+m \right )}\) | \(21\) |
risch | \(\frac {\ln \left (x \right )}{b}-\frac {m \ln \left (x \right )}{b \left (1+m \right )}+\frac {\ln \left (x^{m}+\frac {a}{b x}\right )}{b \left (1+m \right )}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{m + 1} + a\right )}{b {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 18, normalized size = 0.95 \begin {gather*} \frac {\log \left (b x^{m + 1} + a\right )}{b m + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (14) = 28\).
time = 0.39, size = 37, normalized size = 1.95 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge m = -1 \\\frac {x x^{m}}{a \left (m + 1\right )} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: m = -1 \\\frac {\log {\left (\frac {a}{b} + x x^{m} \right )}}{b m + b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left ({\left | b x^{m + 1} + a \right |}\right )}{b m + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 19, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x^{m+1}\right )}{b\,\left (m+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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