Optimal. Leaf size=45 \[ -\frac {a^3}{x}-\frac {9 a^2 b}{4 x^{4/3}}-\frac {9 a b^2}{5 x^{5/3}}-\frac {b^3}{2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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} \[ -\frac {9 a^2 b}{4 x^{4/3}}-\frac {a^3}{x}-\frac {9 a b^2}{5 x^{5/3}}-\frac {b^3}{2 x^2} \]
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 )^3}{x^2} \, dx &=\int \frac {\left (b+a \sqrt [3]{x}\right )^3}{x^3} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {(b+a x)^3}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {b^3}{x^7}+\frac {3 a b^2}{x^6}+\frac {3 a^2 b}{x^5}+\frac {a^3}{x^4}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {b^3}{2 x^2}-\frac {9 a b^2}{5 x^{5/3}}-\frac {9 a^2 b}{4 x^{4/3}}-\frac {a^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \[ -\frac {20 a^3 x+45 a^2 b x^{2/3}+36 a b^2 \sqrt [3]{x}+10 b^3}{20 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 35, normalized size = 0.78 \[ -\frac {20 \, a^{3} x + 45 \, a^{2} b x^{\frac {2}{3}} + 36 \, a b^{2} x^{\frac {1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 35, normalized size = 0.78 \[ -\frac {20 \, a^{3} x + 45 \, a^{2} b x^{\frac {2}{3}} + 36 \, a b^{2} x^{\frac {1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \[ -\frac {a^{3}}{x}-\frac {9 a^{2} b}{4 x^{\frac {4}{3}}}-\frac {9 a \,b^{2}}{5 x^{\frac {5}{3}}}-\frac {b^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 47, normalized size = 1.04 \[ -\frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6}}{2 \, b^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a}{5 \, b^{3}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{2}}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 35, normalized size = 0.78 \[ -\frac {20\,a^3\,x+10\,b^3+36\,a\,b^2\,x^{1/3}+45\,a^2\,b\,x^{2/3}}{20\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 41, normalized size = 0.91 \[ - \frac {a^{3}}{x} - \frac {9 a^{2} b}{4 x^{\frac {4}{3}}} - \frac {9 a b^{2}}{5 x^{\frac {5}{3}}} - \frac {b^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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