Optimal. Leaf size=74 \[ \frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {3 b x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {424, 12, 372,
371} \begin {gather*} \frac {3 b x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 371
Rule 372
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {\int \frac {6 a b^2 x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+(3 b) \int \frac {x^3}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {\left (3 b \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {x^3}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {3 b x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 10.03, size = 62, normalized size = 0.84 \begin {gather*} \frac {5 a x+b x^4-3 a x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{2 \left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {5}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {5}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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