Integrand size = 13, antiderivative size = 34 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=-\frac {b^3}{x}+3 a b^2 x+a^2 b x^3+\frac {a^3 x^5}{5} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 276} \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^3 x^5}{5}+a^2 b x^3+3 a b^2 x-\frac {b^3}{x} \]
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Rule 269
Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+a x^2\right )^3}{x^2} \, dx \\ & = \int \left (3 a b^2+\frac {b^3}{x^2}+3 a^2 b x^2+a^3 x^4\right ) \, dx \\ & = -\frac {b^3}{x}+3 a b^2 x+a^2 b x^3+\frac {a^3 x^5}{5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=-\frac {b^3}{x}+3 a b^2 x+a^2 b x^3+\frac {a^3 x^5}{5} \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {b^{3}}{x}+3 a \,b^{2} x +a^{2} b \,x^{3}+\frac {a^{3} x^{5}}{5}\) | \(33\) |
risch | \(-\frac {b^{3}}{x}+3 a \,b^{2} x +a^{2} b \,x^{3}+\frac {a^{3} x^{5}}{5}\) | \(33\) |
gosper | \(\frac {x^{6} a^{3}+5 a^{2} b \,x^{4}+15 a \,b^{2} x^{2}-5 b^{3}}{5 x}\) | \(37\) |
parallelrisch | \(\frac {x^{6} a^{3}+5 a^{2} b \,x^{4}+15 a \,b^{2} x^{2}-5 b^{3}}{5 x}\) | \(37\) |
norman | \(\frac {a^{2} b \,x^{8}-b^{3} x^{4}+\frac {1}{5} x^{10} a^{3}+3 a \,b^{2} x^{6}}{x^{5}}\) | \(39\) |
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none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^{3} x^{6} + 5 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} - 5 \, b^{3}}{5 \, x} \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^{3} x^{5}}{5} + a^{2} b x^{3} + 3 a b^{2} x - \frac {b^{3}}{x} \]
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none
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac {b^{3}}{x} \]
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none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac {b^{3}}{x} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^3\,x^5}{5}-\frac {b^3}{x}+a^2\,b\,x^3+3\,a\,b^2\,x \]
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