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 = 0.01, 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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(34)=68\).
time = 3.43, size = 96, normalized size = 2.82 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (3 a^3 \left (1-\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^2 b x \left (3-2 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+5 a b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+4 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{28 b^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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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 = 0.26, 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 = 0.32, 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 = 0.59, 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 = 0.00, size = 98, normalized size = 2.88 \begin {gather*} \frac {\frac {3 b \left (\frac {1}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2}-\frac {1}{2} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a+\left (a+b x\right )^{\frac {1}{3}} a^{2}\right )}{b^{2}}+\frac {3 a \left (\frac {1}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )-a \left (a+b x\right )^{\frac {1}{3}}\right )}{b}}{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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