Optimal. Leaf size=38 \[ -\frac {a \left (a+b x^3\right )^{4/3}}{4 b^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {\left (a+b x^3\right )^{7/3}}{7 b^2}-\frac {a \left (a+b x^3\right )^{4/3}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^5 \sqrt [3]{a+b x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int x \sqrt [3]{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a \sqrt [3]{a+b x}}{b}+\frac {(a+b x)^{4/3}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (a+b x^3\right )^{4/3}}{4 b^2}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-3 a^2+a b x^3+4 b^2 x^6\right )}{28 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 25, normalized size = 0.66
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-4 b \,x^{3}+3 a \right )}{28 b^{2}}\) | \(25\) |
trager | \(-\frac {\left (-4 b^{2} x^{6}-a b \,x^{3}+3 a^{2}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{28 b^{2}}\) | \(36\) |
risch | \(-\frac {\left (-4 b^{2} x^{6}-a b \,x^{3}+3 a^{2}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{28 b^{2}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {7}{3}}}{7 \, b^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} a}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, b^{2} x^{6} + a b x^{3} - 3 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{28 \, b^{2}} \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. 63 vs.
\(2 (31) = 62\).
time = 0.14, size = 63, normalized size = 1.66 \begin {gather*} \begin {cases} - \frac {3 a^{2} \sqrt [3]{a + b x^{3}}}{28 b^{2}} + \frac {a x^{3} \sqrt [3]{a + b x^{3}}}{28 b} + \frac {x^{6} \sqrt [3]{a + b x^{3}}}{7} & \text {for}\: b \neq 0 \\\frac {\sqrt [3]{a} x^{6}}{6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 29, normalized size = 0.76 \begin {gather*} \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a}{28 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 33, normalized size = 0.87 \begin {gather*} {\left (b\,x^3+a\right )}^{1/3}\,\left (\frac {x^6}{7}-\frac {3\,a^2}{28\,b^2}+\frac {a\,x^3}{28\,b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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