Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{4/3}}{4 b^2}+\frac {3 (a+b x)^{7/3}}{7 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {3 (a+b x)^{7/3}}{7 b^2}-\frac {3 a (a+b x)^{4/3}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x \sqrt [3]{a+b x} \, dx &=\int \left (-\frac {a \sqrt [3]{a+b x}}{b}+\frac {(a+b x)^{4/3}}{b}\right ) \, dx\\ &=-\frac {3 a (a+b x)^{4/3}}{4 b^2}+\frac {3 (a+b x)^{7/3}}{7 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (-3 a^2+a b x+4 b^2 x^2\right )}{28 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (-4 b x +3 a \right )}{28 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{2}}\) | \(26\) |
trager | \(-\frac {3 \left (-4 x^{2} b^{2}-a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{28 b^{2}}\) | \(32\) |
risch | \(-\frac {3 \left (-4 x^{2} b^{2}-a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{28 b^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b^{2}} - \frac {3 \, {\left (b x + 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 = 0.47, size = 30, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left (4 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )} {\left (b x + 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. 202 vs.
\(2 (31) = 62\).
time = 0.54, size = 202, normalized size = 5.94 \begin {gather*} - \frac {9 a^{\frac {13}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {9 a^{\frac {13}{3}}}{28 a^{2} b^{2} + 28 a b^{3} x} - \frac {6 a^{\frac {10}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {9 a^{\frac {10}{3}} b x}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {15 a^{\frac {7}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {12 a^{\frac {4}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs.
\(2 (26) = 52\).
time = 1.08, size = 67, normalized size = 1.97 \begin {gather*} \frac {3 \, {\left (\frac {7 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a}{b} + \frac {2 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )}}{b}\right )}}{28 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {21\,a\,{\left (a+b\,x\right )}^{4/3}-12\,{\left (a+b\,x\right )}^{7/3}}{28\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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