Optimal. Leaf size=34 \[ -\frac {a^2}{2 x^2}-\frac {6 a b}{7 x^{7/3}}-\frac {3 b^2}{8 x^{8/3}} \]
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Rubi [A] time = 0.02, antiderivative size = 34, 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} \[ -\frac {a^2}{2 x^2}-\frac {6 a b}{7 x^{7/3}}-\frac {3 b^2}{8 x^{8/3}} \]
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^3} \, dx &=\int \frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{11/3}} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {(b+a x)^2}{x^9} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {b^2}{x^9}+\frac {2 a b}{x^8}+\frac {a^2}{x^7}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^2}{8 x^{8/3}}-\frac {6 a b}{7 x^{7/3}}-\frac {a^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.00 \[ -\frac {a^2}{2 x^2}-\frac {6 a b}{7 x^{7/3}}-\frac {3 b^2}{8 x^{8/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 27, normalized size = 0.79 \[ -\frac {28 \, a^{2} x + 48 \, a b x^{\frac {2}{3}} + 21 \, b^{2} x^{\frac {1}{3}}}{56 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 0.76 \[ -\frac {28 \, a^{2} x^{\frac {2}{3}} + 48 \, a b x^{\frac {1}{3}} + 21 \, b^{2}}{56 \, x^{\frac {8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.74 \[ -\frac {a^{2}}{2 x^{2}}-\frac {6 a b}{7 x^{\frac {7}{3}}}-\frac {3 b^{2}}{8 x^{\frac {8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 97, normalized size = 2.85 \[ -\frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8}}{8 \, b^{6}} + \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a}{7 \, b^{6}} - \frac {5 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{2}}{b^{6}} + \frac {6 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{3}}{b^{6}} - \frac {15 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{4}}{4 \, b^{6}} + \frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{5}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 24, normalized size = 0.71 \[ -\frac {a^2}{2\,x^2}-\frac {3\,b^2}{8\,x^{8/3}}-\frac {6\,a\,b}{7\,x^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.85, size = 32, normalized size = 0.94 \[ - \frac {a^{2}}{2 x^{2}} - \frac {6 a b}{7 x^{\frac {7}{3}}} - \frac {3 b^{2}}{8 x^{\frac {8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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