Optimal. Leaf size=17 \[ \frac {2 x \sqrt {b x^n}}{2+n} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30}
\begin {gather*} \frac {2 x \sqrt {b x^n}}{n+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \sqrt {b x^n} \, dx &=\left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{n/2} \, dx\\ &=\frac {2 x \sqrt {b x^n}}{2+n}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {b x^n}}{2+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 16, normalized size = 0.94
method | result | size |
gosper | \(\frac {2 x \sqrt {b \,x^{n}}}{2+n}\) | \(16\) |
risch | \(\frac {2 b x \,x^{n}}{\left (2+n \right ) \sqrt {b \,x^{n}}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {2 \, \sqrt {b x^{n}} x}{n + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {2 x \sqrt {b x^{n}}}{n + 2} & \text {for}\: n \neq -2 \\\int \sqrt {\frac {b}{x^{2}}}\, dx & \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 = 1.48, size = 15, normalized size = 0.88 \begin {gather*} \frac {2\,x\,\sqrt {b\,x^n}}{n+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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