Optimal. Leaf size=19 \[ \frac {2 x^3 \sqrt {b x^n}}{n+6} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac {2 x^3 \sqrt {b x^n}}{n+6} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int x^2 \sqrt {b x^n} \, dx &=\left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{2+\frac {n}{2}} \, dx\\ &=\frac {2 x^3 \sqrt {b x^n}}{6+n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {2 x^3 \sqrt {b x^n}}{n+6} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {b x^{n}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.95 \[ \frac {2 \sqrt {b \,x^{n}}\, x^{3}}{n +6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 17, normalized size = 0.89 \[ \frac {2 \, \sqrt {b x^{n}} x^{3}}{n + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 17, normalized size = 0.89 \[ \frac {2\,x^3\,\sqrt {b\,x^n}}{n+6} \]
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 \[ \begin {cases} \frac {2 \sqrt {b} x^{3} \sqrt {x^{n}}}{n + 6} & \text {for}\: n \neq -6 \\\int x^{2} \sqrt {\frac {b}{x^{6}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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