Optimal. Leaf size=28 \[ \frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45}
\begin {gather*} \frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{a+b x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.96 \begin {gather*} \frac {b x^n-a \log \left (b n \left (a+b x^n\right )\right )}{b^2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 31, normalized size = 1.11
method | result | size |
risch | \(\frac {x^{n}}{b n}-\frac {a \ln \left (x^{n}+\frac {a}{b}\right )}{b^{2} n}\) | \(31\) |
norman | \(\frac {{\mathrm e}^{n \ln \left (x \right )}}{b n}-\frac {a \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{b^{2} n}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 32, normalized size = 1.14 \begin {gather*} \frac {x^{n}}{b n} - \frac {a \log \left (\frac {b x^{n} + a}{b}\right )}{b^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 24, normalized size = 0.86 \begin {gather*} \frac {b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} 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. 41 vs.
\(2 (20) = 40\).
time = 4.59, size = 41, normalized size = 1.46 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{2 n}}{2 a n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\- \frac {a \log {\left (\frac {a}{b} + x^{n} \right )}}{b^{2} n} + \frac {x^{n}}{b 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 [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^{2\,n-1}}{a+b\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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