Integrand size = 13, antiderivative size = 41 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=-\frac {a^3}{4 x^4}-\frac {3 a^2 b}{x}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^5}{5} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=-\frac {a^3}{4 x^4}-\frac {3 a^2 b}{x}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^5}{5} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3}{x^5}+\frac {3 a^2 b}{x^2}+3 a b^2 x+b^3 x^4\right ) \, dx \\ & = -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{x}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^5}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=-\frac {a^3}{4 x^4}-\frac {3 a^2 b}{x}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^5}{5} \]
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Time = 3.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{3}}{4 x^{4}}-\frac {3 a^{2} b}{x}+\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{5}}{5}\) | \(36\) |
norman | \(\frac {\frac {1}{5} b^{3} x^{9}+\frac {3}{2} a \,b^{2} x^{6}-3 a^{2} b \,x^{3}-\frac {1}{4} a^{3}}{x^{4}}\) | \(37\) |
gosper | \(-\frac {-4 b^{3} x^{9}-30 a \,b^{2} x^{6}+60 a^{2} b \,x^{3}+5 a^{3}}{20 x^{4}}\) | \(38\) |
risch | \(\frac {b^{3} x^{5}}{5}+\frac {3 a \,b^{2} x^{2}}{2}+\frac {-3 a^{2} b \,x^{3}-\frac {1}{4} a^{3}}{x^{4}}\) | \(38\) |
parallelrisch | \(\frac {4 b^{3} x^{9}+30 a \,b^{2} x^{6}-60 a^{2} b \,x^{3}-5 a^{3}}{20 x^{4}}\) | \(38\) |
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=\frac {4 \, b^{3} x^{9} + 30 \, a b^{2} x^{6} - 60 \, a^{2} b x^{3} - 5 \, a^{3}}{20 \, x^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=\frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{5}}{5} + \frac {- a^{3} - 12 a^{2} b x^{3}}{4 x^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{2} \, a b^{2} x^{2} - \frac {12 \, a^{2} b x^{3} + a^{3}}{4 \, x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=\frac {1}{5} \, b^{3} x^{5} + \frac {3}{2} \, a b^{2} x^{2} - \frac {12 \, a^{2} b x^{3} + a^{3}}{4 \, x^{4}} \]
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Time = 5.40 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^3}{x^5} \, dx=\frac {b^3\,x^5}{5}-\frac {\frac {a^3}{4}+3\,b\,a^2\,x^3}{x^4}+\frac {3\,a\,b^2\,x^2}{2} \]
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