Optimal. Leaf size=59 \[ \frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\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,x,x^2\right )\\ &=\frac {3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}-\frac {3 a \left (a+b x^2\right )^{4/3}}{4 b^3}+\frac {3 \left (a+b x^2\right )^{7/3}}{14 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.66 \begin {gather*} \frac {3 \sqrt [3]{a+b x^2} \left (9 a^2-3 a b x^2+2 b^2 x^4\right )}{28 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 36, normalized size = 0.61
method | result | size |
gosper | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\) | \(36\) |
trager | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\) | \(36\) |
risch | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{4}-3 a b \,x^{2}+9 a^{2}\right )}{28 b^{3}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}}}{14 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a}{4 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{2}}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 35, normalized size = 0.59 \begin {gather*} \frac {3 \, {\left (2 \, b^{2} x^{4} - 3 \, a b x^{2} + 9 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{28 \, b^{3}} \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. 631 vs.
\(2 (54) = 108\).
time = 0.87, size = 631, normalized size = 10.69 \begin {gather*} \frac {27 a^{\frac {31}{3}} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {31}{3}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {72 a^{\frac {28}{3}} b x^{2} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {28}{3}} b x^{2}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {60 a^{\frac {25}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {81 a^{\frac {25}{3}} b^{2} x^{4}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {18 a^{\frac {22}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac {27 a^{\frac {22}{3}} b^{3} x^{6}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {9 a^{\frac {19}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac {6 a^{\frac {16}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac {b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 47, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{2}}{2 \, b^{3}} + \frac {3 \, {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} a\right )}}{28 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.77, size = 36, normalized size = 0.61 \begin {gather*} {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {27\,a^2}{28\,b^3}+\frac {3\,x^4}{14\,b}-\frac {9\,a\,x^2}{28\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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