Optimal. Leaf size=32 \[ \frac {3 a}{b^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 b^2} \]
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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^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac {a}{b (a+b x)^{4/3}}+\frac {1}{b \sqrt [3]{a+b x}}\right ) \, dx\\ &=\frac {3 a}{b^2 \sqrt [3]{a+b x}}+\frac {3 (a+b x)^{2/3}}{2 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)}{2 b^2 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 25, normalized size = 0.78
method | result | size |
gosper | \(\frac {\frac {3 b x}{2}+\frac {9 a}{2}}{\left (b x +a \right )^{\frac {1}{3}} b^{2}}\) | \(20\) |
trager | \(\frac {\frac {3 b x}{2}+\frac {9 a}{2}}{\left (b x +a \right )^{\frac {1}{3}} b^{2}}\) | \(20\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{2}}\) | \(25\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{2}}\) | \(25\) |
risch | \(\frac {3 a}{b^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.81 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}}}{2 \, b^{2}} + \frac {3 \, a}{{\left (b x + a\right )}^{\frac {1}{3}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 29, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (b x + 3 \, a\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{2 \, {\left (b^{3} x + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 41, normalized size = 1.28 \begin {gather*} \begin {cases} \frac {9 a}{2 b^{2} \sqrt [3]{a + b x}} + \frac {3 x}{2 b \sqrt [3]{a + b x}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {4}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 30, normalized size = 0.94 \begin {gather*} \frac {3 \, {\left (\frac {{\left (b x + a\right )}^{\frac {2}{3}}}{b} + \frac {2 \, a}{{\left (b x + a\right )}^{\frac {1}{3}} b}\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 0.62 \begin {gather*} \frac {9\,a+3\,b\,x}{2\,b^2\,{\left (a+b\,x\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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