Optimal. Leaf size=77 \[ \frac {9 b \sqrt [3]{a x^3+b x^6}}{4 a^3 x^2}-\frac {3 \sqrt [3]{a x^3+b x^6}}{4 a^2 x^5}+\frac {1}{2 a x^2 \left (a x^3+b x^6\right )^{2/3}} \]
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Rubi [A] time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2001, 2016, 2000} \[ \frac {9 b \sqrt [3]{a x^3+b x^6}}{4 a^3 x^2}-\frac {3 \sqrt [3]{a x^3+b x^6}}{4 a^2 x^5}+\frac {1}{2 a x^2 \left (a x^3+b x^6\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2000
Rule 2001
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{\left (a x^3+b x^6\right )^{5/3}} \, dx &=\frac {1}{2 a x^2 \left (a x^3+b x^6\right )^{2/3}}+\frac {3 \int \frac {1}{x^3 \left (a x^3+b x^6\right )^{2/3}} \, dx}{a}\\ &=\frac {1}{2 a x^2 \left (a x^3+b x^6\right )^{2/3}}-\frac {3 \sqrt [3]{a x^3+b x^6}}{4 a^2 x^5}-\frac {(9 b) \int \frac {1}{\left (a x^3+b x^6\right )^{2/3}} \, dx}{4 a^2}\\ &=\frac {1}{2 a x^2 \left (a x^3+b x^6\right )^{2/3}}-\frac {3 \sqrt [3]{a x^3+b x^6}}{4 a^2 x^5}+\frac {9 b \sqrt [3]{a x^3+b x^6}}{4 a^3 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.60 \[ \frac {-a^2+6 a b x^3+9 b^2 x^6}{4 a^3 x^2 \left (x^3 \left (a+b x^3\right )\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 54, normalized size = 0.70 \[ \frac {{\left (9 \, b^{2} x^{6} + 6 \, a b x^{3} - a^{2}\right )} {\left (b x^{6} + a x^{3}\right )}^{\frac {1}{3}}}{4 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 23.79, size = 52, normalized size = 0.68 \[ \frac {b^{2}}{2 \, a^{3} {\left (b + \frac {a}{x^{3}}\right )}^{\frac {2}{3}}} - \frac {a^{9} {\left (b + \frac {a}{x^{3}}\right )}^{\frac {4}{3}} - 8 \, a^{9} {\left (b + \frac {a}{x^{3}}\right )}^{\frac {1}{3}} b}{4 \, a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.60 \[ -\frac {\left (b \,x^{3}+a \right ) \left (-9 b^{2} x^{6}-6 b \,x^{3} a +a^{2}\right ) x}{4 \left (b \,x^{6}+a \,x^{3}\right )^{\frac {5}{3}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 38, normalized size = 0.49 \[ \frac {9 \, b^{2} x^{6} + 6 \, a b x^{3} - a^{2}}{4 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 51, normalized size = 0.66 \[ \frac {{\left (b\,x^6+a\,x^3\right )}^{1/3}\,\left (-a^2+6\,a\,b\,x^3+9\,b^2\,x^6\right )}{4\,a^3\,x^5\,\left (b\,x^3+a\right )} \]
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 \[ \int \frac {1}{\left (a x^{3} + b x^{6}\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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