Optimal. Leaf size=32 \[ -\frac {3 a \sqrt [3]{a+b x}}{b^2}+\frac {3 (a+b x)^{4/3}}{4 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, 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)^{4/3}}{4 b^2}-\frac {3 a \sqrt [3]{a+b x}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^{2/3}} \, dx &=\int \left (-\frac {a}{b (a+b x)^{2/3}}+\frac {\sqrt [3]{a+b x}}{b}\right ) \, dx\\ &=-\frac {3 a \sqrt [3]{a+b x}}{b^2}+\frac {3 (a+b x)^{4/3}}{4 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.72 \begin {gather*} \frac {3 (-3 a+b x) \sqrt [3]{a+b x}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 26, normalized size = 0.81
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
trager | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
risch | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +3 a \right )}{4 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a \left (b x +a \right )^{\frac {1}{3}}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a \left (b x +a \right )^{\frac {1}{3}}}{b^{2}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 0.81 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}}}{4 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 19, normalized size = 0.59 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (b x - 3 \, a\right )}}{4 \, 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. 162 vs.
\(2 (29) = 58\).
time = 0.52, size = 162, normalized size = 5.06 \begin {gather*} - \frac {9 a^{\frac {10}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {9 a^{\frac {10}{3}}}{4 a^{2} b^{2} + 4 a b^{3} x} - \frac {6 a^{\frac {7}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {9 a^{\frac {7}{3}} b x}{4 a^{2} b^{2} + 4 a b^{3} x} + \frac {3 a^{\frac {4}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{4 a^{2} b^{2} + 4 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 23, normalized size = 0.72 \begin {gather*} \frac {3 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )}}{4 \, b^{2}} \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 = 0.78 \begin {gather*} -\frac {12\,a\,{\left (a+b\,x\right )}^{1/3}-3\,{\left (a+b\,x\right )}^{4/3}}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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