Optimal. Leaf size=56 \[ \frac {a^2 \sqrt [3]{a+b x^3}}{b^3}-\frac {a \left (a+b x^3\right )^{4/3}}{2 b^3}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, 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 {a^2 \sqrt [3]{a+b x^3}}{b^3}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3}-\frac {a \left (a+b x^3\right )^{4/3}}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \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^3\right )\\ &=\frac {a^2 \sqrt [3]{a+b x^3}}{b^3}-\frac {a \left (a+b x^3\right )^{4/3}}{2 b^3}+\frac {\left (a+b x^3\right )^{7/3}}{7 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.70 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (9 a^2-3 a b x^3+2 b^2 x^6\right )}{14 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 36, normalized size = 0.64
method | result | size |
gosper | \(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{6}-3 a b \,x^{3}+9 a^{2}\right )}{14 b^{3}}\) | \(36\) |
trager | \(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{6}-3 a b \,x^{3}+9 a^{2}\right )}{14 b^{3}}\) | \(36\) |
risch | \(\frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (2 b^{2} x^{6}-3 a b \,x^{3}+9 a^{2}\right )}{14 b^{3}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 46, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {7}{3}}}{7 \, b^{3}} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} a}{2 \, b^{3}} + \frac {{\left (b x^{3} + 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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Fricas [A]
time = 0.36, size = 35, normalized size = 0.62 \begin {gather*} \frac {{\left (2 \, b^{2} x^{6} - 3 \, a b x^{3} + 9 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{14 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 68, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {9 a^{2} \sqrt [3]{a + b x^{3}}}{14 b^{3}} - \frac {3 a x^{3} \sqrt [3]{a + b x^{3}}}{14 b^{2}} + \frac {x^{6} \sqrt [3]{a + b x^{3}}}{7 b} & \text {for}\: b \neq 0 \\\frac {x^{9}}{9 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.87, size = 46, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2}}{b^{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a}{14 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 36, normalized size = 0.64 \begin {gather*} {\left (b\,x^3+a\right )}^{1/3}\,\left (\frac {9\,a^2}{14\,b^3}+\frac {x^6}{7\,b}-\frac {3\,a\,x^3}{14\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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