Optimal. Leaf size=51 \[ \frac {3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3} \]
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Rubi [A] time = 0.01, antiderivative size = 51, 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 = {43} \begin {gather*} \frac {3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^{2/3}} \, dx &=\int \left (\frac {a^2}{b^2 (a+b x)^{2/3}}-\frac {2 a \sqrt [3]{a+b x}}{b^2}+\frac {(a+b x)^{4/3}}{b^2}\right ) \, dx\\ &=\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{2 b^3}+\frac {3 (a+b x)^{7/3}}{7 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.69 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (9 a^2-3 a b x+2 b^2 x^2\right )}{14 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 45, normalized size = 0.88 \begin {gather*} \frac {3 \left (14 a^2 \sqrt [3]{a+b x}+2 (a+b x)^{7/3}-7 a (a+b x)^{4/3}\right )}{14 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 31, normalized size = 0.61 \begin {gather*} \frac {3 \, {\left (2 \, b^{2} x^{2} - 3 \, a b x + 9 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{14 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 37, normalized size = 0.73 \begin {gather*} \frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )}}{14 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.63 \begin {gather*} \frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (2 b^{2} x^{2}-3 a b x +9 a^{2}\right )}{14 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 41, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a}{2 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.73 \begin {gather*} \frac {6\,{\left (a+b\,x\right )}^{7/3}-21\,a\,{\left (a+b\,x\right )}^{4/3}+42\,a^2\,{\left (a+b\,x\right )}^{1/3}}{14\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.82, size = 600, normalized size = 11.76 \begin {gather*} \frac {27 a^{\frac {31}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {31}{3}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {28}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {28}{3}} b x}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {25}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {25}{3}} b^{2} x^{2}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {18 a^{\frac {22}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {22}{3}} b^{3} x^{3}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {9 a^{\frac {19}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac {6 a^{\frac {16}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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