Optimal. Leaf size=19 \[ \frac {x^4 \left (b x^2\right )^p}{2 (2+p)} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30}
\begin {gather*} \frac {x^4 \left (b x^2\right )^p}{2 (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int x^3 \left (b x^2\right )^p \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{3+2 p} \, dx\\ &=\frac {x^4 \left (b x^2\right )^p}{2 (2+p)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.95 \begin {gather*} \frac {x^4 \left (b x^2\right )^p}{4+2 p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.95
method | result | size |
gosper | \(\frac {\left (b \,x^{2}\right )^{p} x^{4}}{4+2 p}\) | \(18\) |
risch | \(\frac {\left (b \,x^{2}\right )^{p} x^{4}}{4+2 p}\) | \(18\) |
norman | \(\frac {x^{4} {\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{4+2 p}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.95 \begin {gather*} \frac {b^{p} {\left (x^{2}\right )}^{p} x^{4}}{2 \, {\left (p + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 0.89 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x^{4}}{2 \, {\left (p + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 22, normalized size = 1.16 \begin {gather*} \begin {cases} \frac {x^{4} \left (b x^{2}\right )^{p}}{2 p + 4} & \text {for}\: p \neq -2 \\\frac {\log {\left (x \right )}}{b^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 17, normalized size = 0.89 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x^{4}}{2 \, {\left (p + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 18, normalized size = 0.95 \begin {gather*} \frac {x^4\,{\left (b\,x^2\right )}^p}{2\,\left (p+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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