Optimal. Leaf size=15 \[ \frac {\log \left (b+a x^n\right )}{a n} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1607, 266}
\begin {gather*} \frac {\log \left (a x^n+b\right )}{a n} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{a x+b x^{1-n}} \, dx &=\int \frac {x^{-1+n}}{b+a x^n} \, dx\\ &=\frac {\log \left (b+a x^n\right )}{a n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} \frac {\log \left (b n+a n x^n\right )}{a n} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs.
\(2(15)=30\).
time = 0.40, size = 37, normalized size = 2.47
method | result | size |
norman | \(\frac {\left (-1+n \right ) \ln \left (x \right )}{a n}+\frac {\ln \left (a x +b \,{\mathrm e}^{\left (1-n \right ) \ln \left (x \right )}\right )}{a n}\) | \(37\) |
risch | \(-\frac {\ln \left (x \right )}{a n}+\frac {\ln \left (x \right )}{a}+\frac {\ln \left (x^{1-n}+\frac {a x}{b}\right )}{a n}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 19, normalized size = 1.27 \begin {gather*} \frac {\log \left (\frac {a x^{n} + b}{a}\right )}{a n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.08, size = 28, normalized size = 1.87 \begin {gather*} \frac {{\left (n - 1\right )} \log \left (x\right ) + \log \left (a x + b x^{-n + 1}\right )}{a n} \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. 39 vs.
\(2 (10) = 20\).
time = 0.67, size = 39, normalized size = 2.60 \begin {gather*} \begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {x^{n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} + \frac {\log {\left (\frac {a}{b} + x^{- n} \right )}}{a n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.22, size = 34, normalized size = 2.27 \begin {gather*} \frac {\ln \left (a\,x+b\,x^{1-n}\right )}{a\,n}+\frac {\ln \left (x\right )\,\left (n-1\right )}{a\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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