Optimal. Leaf size=20 \[ \frac {2 b x^n \sqrt {b x^n}}{3 n} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, 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}
\begin {gather*} \frac {2 b x^n \sqrt {b x^n}}{3 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {\left (b x^n\right )^{3/2}}{x} \, dx &=\left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{-1+\frac {3 n}{2}} \, dx\\ &=\frac {2 b x^n \sqrt {b x^n}}{3 n}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \left (b x^n\right )^{3/2}}{3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 13, normalized size = 0.65
method | result | size |
gosper | \(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\) | \(13\) |
derivativedivides | \(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\) | \(13\) |
default | \(\frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}}}{3 n}\) | \(13\) |
risch | \(\frac {2 b^{2} x^{2 n}}{3 n \sqrt {b \,x^{n}}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 12, normalized size = 0.60 \begin {gather*} \frac {2 \, \left (b x^{n}\right )^{\frac {3}{2}}}{3 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {b x^{n}} b x^{n}}{3 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.56, size = 20, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {2 \left (b x^{n}\right )^{\frac {3}{2}}}{3 n} & \text {for}\: n \neq 0 \\b^{\frac {3}{2}} \log {\left (x \right )} & \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 = 0.93, size = 16, normalized size = 0.80 \begin {gather*} \frac {2\,b\,x^n\,\sqrt {b\,x^n}}{3\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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