Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{7/3}}{7 b^2}+\frac {3 (a+b x)^{10/3}}{10 b^2} \]
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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)^{10/3}}{10 b^2}-\frac {3 a (a+b x)^{7/3}}{7 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x (a+b x)^{4/3} \, dx &=\int \left (-\frac {a (a+b x)^{4/3}}{b}+\frac {(a+b x)^{7/3}}{b}\right ) \, dx\\ &=-\frac {3 a (a+b x)^{7/3}}{7 b^2}+\frac {3 (a+b x)^{10/3}}{10 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.71 \begin {gather*} \frac {3 (a+b x)^{7/3} (-3 a+7 b x)}{70 b^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 2.07, size = 53, normalized size = 1.56 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {3 \left (-3 a^3+a^2 b x+b^2 x^2 \left (11 a+7 b x\right )\right ) \left (a+b x\right )^{\frac {1}{3}}}{70 b^2},b\text {!=}0\right \}\right \},\frac {a^{\frac {4}{3}} x^2}{2}\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 | \(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-7 b x +3 a \right )}{70 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\) | \(26\) |
trager | \(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\) | \(43\) |
risch | \(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\) | \(43\) |
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 {10}{3}}}{10 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 41, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (7 \, b^{3} x^{3} + 11 \, a b^{2} x^{2} + a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{70 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 80, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {9 a^{3} \sqrt [3]{a + b x}}{70 b^{2}} + \frac {3 a^{2} x \sqrt [3]{a + b x}}{70 b} + \frac {33 a x^{2} \sqrt [3]{a + b x}}{70} + \frac {3 b x^{3} \sqrt [3]{a + b x}}{10} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{2}}{2} & \text {otherwise} \end {cases} \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. 118 vs.
\(2 (26) = 52\).
time = 0.00, size = 187, normalized size = 5.50 \begin {gather*} \frac {\frac {3 b^{2} \left (\frac {1}{10} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3}-\frac {3}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a+\frac {3}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{2}-\left (a+b x\right )^{\frac {1}{3}} a^{3}\right )}{b^{3}}+\frac {6 a 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^{2} \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 {30\,a\,{\left (a+b\,x\right )}^{7/3}-21\,{\left (a+b\,x\right )}^{10/3}}{70\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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