Optimal. Leaf size=32 \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {263, 266, 43} \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^2}{x^2} \, dx &=\int \frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{8/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {b^2}{x^6}+\frac {2 a b}{x^5}+\frac {a^2}{x^4}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^2}{5 x^{5/3}}-\frac {3 a b}{2 x^{4/3}}-\frac {a^2}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \[ -\frac {a^2}{x}-\frac {3 a b}{2 x^{4/3}}-\frac {3 b^2}{5 x^{5/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 27, normalized size = 0.84 \[ -\frac {10 \, a^{2} x + 15 \, a b x^{\frac {2}{3}} + 6 \, b^{2} x^{\frac {1}{3}}}{10 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 26, normalized size = 0.81 \[ -\frac {10 \, a^{2} x^{\frac {2}{3}} + 15 \, a b x^{\frac {1}{3}} + 6 \, b^{2}}{10 \, x^{\frac {5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.78 \[ -\frac {a^{2}}{x}-\frac {3 a b}{2 x^{\frac {4}{3}}}-\frac {3 b^{2}}{5 x^{\frac {5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 47, normalized size = 1.47 \[ -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5}}{5 \, b^{3}} + \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a}{2 \, b^{3}} - \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.75 \[ -\frac {a^2}{x}-\frac {3\,b^2}{5\,x^{5/3}}-\frac {3\,a\,b}{2\,x^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 29, normalized size = 0.91 \[ - \frac {a^{2}}{x} - \frac {3 a b}{2 x^{\frac {4}{3}}} - \frac {3 b^{2}}{5 x^{\frac {5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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