Optimal. Leaf size=49 \[ -\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^{4/3}} \, dx &=\int \left (\frac {a^2}{b^2 (a+b x)^{4/3}}-\frac {2 a}{b^2 \sqrt [3]{a+b x}}+\frac {(a+b x)^{2/3}}{b^2}\right ) \, dx\\ &=-\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {3 (a+b x)^{5/3}}{5 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.69 \begin {gather*} \frac {3 \left (-9 a^2-3 a b x+b^2 x^2\right )}{5 b^3 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 38, normalized size = 0.78
method | result | size |
gosper | \(-\frac {3 \left (-x^{2} b^{2}+3 a b x +9 a^{2}\right )}{5 \left (b x +a \right )^{\frac {1}{3}} b^{3}}\) | \(32\) |
trager | \(-\frac {3 \left (-x^{2} b^{2}+3 a b x +9 a^{2}\right )}{5 \left (b x +a \right )^{\frac {1}{3}} b^{3}}\) | \(32\) |
risch | \(-\frac {3 \left (-b x +4 a \right ) \left (b x +a \right )^{\frac {2}{3}}}{5 b^{3}}-\frac {3 a^{2}}{b^{3} \left (b x +a \right )^{\frac {1}{3}}}\) | \(37\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-3 a \left (b x +a \right )^{\frac {2}{3}}-\frac {3 a^{2}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{3}}\) | \(38\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-3 a \left (b x +a \right )^{\frac {2}{3}}-\frac {3 a^{2}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{3}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 41, normalized size = 0.84 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}}}{5 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{b^{3}} - \frac {3 \, a^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 40, normalized size = 0.82 \begin {gather*} \frac {3 \, {\left (b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{5 \, {\left (b^{4} x + a b^{3}\right )}} \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. 534 vs.
\(2 (46) = 92\).
time = 0.83, size = 534, normalized size = 10.90 \begin {gather*} - \frac {27 a^{\frac {29}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {27 a^{\frac {29}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {63 a^{\frac {26}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {81 a^{\frac {26}{3}} b x}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {42 a^{\frac {23}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {81 a^{\frac {23}{3}} b^{2} x^{2}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac {3 a^{\frac {20}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {27 a^{\frac {20}{3}} b^{3} x^{3}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac {3 a^{\frac {17}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 46, normalized size = 0.94 \begin {gather*} -\frac {3 \, a^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{\frac {5}{3}} b^{12} - 5 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{12}\right )}}{5 \, b^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 35, normalized size = 0.71 \begin {gather*} -\frac {15\,a\,\left (a+b\,x\right )-3\,{\left (a+b\,x\right )}^2+15\,a^2}{5\,b^3\,{\left (a+b\,x\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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