Optimal. Leaf size=47 \[ \frac {3 x}{4 a \sqrt [3]{a+b x^3}}+\frac {x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {378, 191} \[ \frac {3 x}{4 a \sqrt [3]{a+b x^3}}+\frac {x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 378
Rubi steps
\begin {align*} \int \frac {a-b x^3}{\left (a+b x^3\right )^{7/3}} \, dx &=\frac {x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}}+\frac {3}{4} \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx\\ &=\frac {x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}}+\frac {3 x}{4 a \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.60 \[ \frac {x \left (2 a+b x^3\right )}{2 a \left (a+b x^3\right )^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 44, normalized size = 0.94 \[ \frac {{\left (b x^{4} + 2 \, a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, {\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac {7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.53 \[ \frac {\left (b \,x^{3}+2 a \right ) x}{2 \left (b \,x^{3}+a \right )^{\frac {4}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 50, normalized size = 1.06 \[ -\frac {{\left (b - \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{4 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a} - \frac {b x^{4}}{4 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 27, normalized size = 0.57 \[ \frac {x\,\left (b\,x^3+a\right )+a\,x}{2\,a\,{\left (b\,x^3+a\right )}^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 91.68, size = 190, normalized size = 4.04 \[ a \left (\frac {4 a x \Gamma \left (\frac {1}{3}\right )}{9 a^{\frac {10}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right ) + 9 a^{\frac {7}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right )} + \frac {3 b x^{4} \Gamma \left (\frac {1}{3}\right )}{9 a^{\frac {10}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right ) + 9 a^{\frac {7}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right )}\right ) - \frac {b x^{4} \Gamma \left (\frac {4}{3}\right )}{3 a^{\frac {7}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right ) + 3 a^{\frac {4}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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