Integrand size = 15, antiderivative size = 45 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {b^3}{2 x^2}-\frac {9 a b^2}{5 x^{5/3}}-\frac {9 a^2 b}{4 x^{4/3}}-\frac {a^3}{x} \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 45} \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {9 a^2 b}{4 x^{4/3}}-\frac {9 a b^2}{5 x^{5/3}}-\frac {b^3}{2 x^2} \]
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Rule 45
Rule 269
Rule 272
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+a \sqrt [3]{x}\right )^3}{x^3} \, dx \\ & = 3 \text {Subst}\left (\int \frac {(b+a x)^3}{x^7} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {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*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=\frac {-10 b^3-36 a b^2 \sqrt [3]{x}-45 a^2 b x^{2/3}-20 a^3 x}{20 x^2} \]
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Time = 3.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {b^{3}}{2 x^{2}}-\frac {9 a \,b^{2}}{5 x^{\frac {5}{3}}}-\frac {9 a^{2} b}{4 x^{\frac {4}{3}}}-\frac {a^{3}}{x}\) | \(36\) |
default | \(-\frac {b^{3}}{2 x^{2}}-\frac {9 a \,b^{2}}{5 x^{\frac {5}{3}}}-\frac {9 a^{2} b}{4 x^{\frac {4}{3}}}-\frac {a^{3}}{x}\) | \(36\) |
trager | \(\frac {\left (-1+x \right ) \left (2 a^{3} x +b^{3} x +b^{3}\right )}{2 x^{2}}-\frac {9 a \,b^{2}}{5 x^{\frac {5}{3}}}-\frac {9 a^{2} b}{4 x^{\frac {4}{3}}}\) | \(43\) |
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {20 \, a^{3} x + 45 \, a^{2} b x^{\frac {2}{3}} + 36 \, a b^{2} x^{\frac {1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=- \frac {a^{3}}{x} - \frac {9 a^{2} b}{4 x^{\frac {4}{3}}} - \frac {9 a b^{2}}{5 x^{\frac {5}{3}}} - \frac {b^{3}}{2 x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {{\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6}}{2 \, b^{3}} + \frac {6 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a}{5 \, b^{3}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{2}}{4 \, b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {20 \, a^{3} x + 45 \, a^{2} b x^{\frac {2}{3}} + 36 \, a b^{2} x^{\frac {1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \]
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Time = 5.86 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^3}{x^2} \, dx=-\frac {20\,a^3\,x+10\,b^3+36\,a\,b^2\,x^{1/3}+45\,a^2\,b\,x^{2/3}}{20\,x^2} \]
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