Optimal. Leaf size=23 \[ \frac {\left (a+b x^4\right )^{1+p}}{4 b (1+p)} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267}
\begin {gather*} \frac {\left (a+b x^4\right )^{p+1}}{4 b (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int x^3 \left (a+b x^4\right )^p \, dx &=\frac {\left (a+b x^4\right )^{1+p}}{4 b (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^4\right )^{1+p}}{4 b (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 22, normalized size = 0.96
method | result | size |
gosper | \(\frac {\left (b \,x^{4}+a \right )^{1+p}}{4 b \left (1+p \right )}\) | \(22\) |
derivativedivides | \(\frac {\left (b \,x^{4}+a \right )^{1+p}}{4 b \left (1+p \right )}\) | \(22\) |
default | \(\frac {\left (b \,x^{4}+a \right )^{1+p}}{4 b \left (1+p \right )}\) | \(22\) |
risch | \(\frac {\left (b \,x^{4}+a \right ) \left (b \,x^{4}+a \right )^{p}}{4 b \left (1+p \right )}\) | \(27\) |
norman | \(\frac {x^{4} {\mathrm e}^{p \ln \left (b \,x^{4}+a \right )}}{4 p +4}+\frac {a \,{\mathrm e}^{p \ln \left (b \,x^{4}+a \right )}}{4 b \left (1+p \right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 21, normalized size = 0.91 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{p + 1}}{4 \, b {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (b x^{4} + a\right )} {\left (b x^{4} + a\right )}^{p}}{4 \, {\left (b p + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (15) = 30\).
time = 0.53, size = 104, normalized size = 4.52 \begin {gather*} \begin {cases} \frac {x^{4}}{4 a} & \text {for}\: b = 0 \wedge p = -1 \\\frac {a^{p} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\log {\left (x - \sqrt [4]{- \frac {a}{b}} \right )}}{4 b} + \frac {\log {\left (x + \sqrt [4]{- \frac {a}{b}} \right )}}{4 b} + \frac {\log {\left (x^{2} + \sqrt {- \frac {a}{b}} \right )}}{4 b} & \text {for}\: p = -1 \\\frac {a \left (a + b x^{4}\right )^{p}}{4 b p + 4 b} + \frac {b x^{4} \left (a + b x^{4}\right )^{p}}{4 b p + 4 b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 21, normalized size = 0.91 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{p + 1}}{4 \, b {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 21, normalized size = 0.91 \begin {gather*} \frac {{\left (b\,x^4+a\right )}^{p+1}}{4\,b\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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