Optimal. Leaf size=23 \[ \frac {\left (a+b x^n\right )^{1+p}}{b n (1+p)} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, 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 = {267}
\begin {gather*} \frac {\left (a+b x^n\right )^{p+1}}{b n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int x^{-1+n} \left (a+b x^n\right )^p \, dx &=\frac {\left (a+b x^n\right )^{1+p}}{b n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^n\right )^{1+p}}{b n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 29, normalized size = 1.26
method | result | size |
risch | \(\frac {\left (a +b \,x^{n}\right ) \left (a +b \,x^{n}\right )^{p}}{b \left (1+p \right ) n}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (b x^{n} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 27, normalized size = 1.17 \begin {gather*} \frac {{\left (b x^{n} + a\right )} {\left (b x^{n} + a\right )}^{p}}{b n p + b 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. 75 vs.
\(2 (15) = 30\).
time = 21.85, size = 75, normalized size = 3.26 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac {a^{p} x^{n}}{n} & \text {for}\: b = 0 \\\left (a + b\right )^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{b n} & \text {for}\: p = -1 \\\frac {a \left (a + b x^{n}\right )^{p}}{b n p + b n} + \frac {b x^{n} \left (a + b x^{n}\right )^{p}}{b n p + b n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (b x^{n} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.59, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,x^n\right )}^{p+1}}{b\,n\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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