Optimal. Leaf size=23 \[ \frac {2 (a+b x)^{\frac {2+p}{2}}}{b (2+p)} \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {32}
\begin {gather*} \frac {2 (a+b x)^{\frac {p+2}{2}}}{b (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin {align*} \int (a+b x)^{p/2} \, dx &=\frac {2 (a+b x)^{\frac {2+p}{2}}}{b (2+p)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.04 \begin {gather*} \frac {2 (a+b x)^{1+\frac {p}{2}}}{2 b+b p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 25, normalized size = 1.09
method | result | size |
gosper | \(\frac {2 \left (b x +a \right ) \left (b x +a \right )^{\frac {p}{2}}}{b \left (2+p \right )}\) | \(25\) |
risch | \(\frac {2 \left (b x +a \right ) \left (b x +a \right )^{\frac {p}{2}}}{b \left (2+p \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.62, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {1}{2} \, p + 1}}{b {\left (p + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (b x + a\right )} \sqrt {b x + a}^{p}}{b p + 2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 24, normalized size = 1.04 \begin {gather*} \frac {\begin {cases} \frac {\left (a + b x\right )^{\frac {p}{2} + 1}}{\frac {p}{2} + 1} & \text {for}\: p \neq -2 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {1}{2} \, p + 1}}{b {\left (p + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 21, normalized size = 0.91 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{\frac {p}{2}+1}}{b\,\left (p+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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