Integrand size = 13, antiderivative size = 39 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=-\frac {a^3}{3 x^3}+3 a^2 b x+\frac {3}{5} a b^2 x^5+\frac {b^3 x^9}{9} \]
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Time = 0.01 (sec) , antiderivative size = 39, 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} \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=-\frac {a^3}{3 x^3}+3 a^2 b x+\frac {3}{5} a b^2 x^5+\frac {b^3 x^9}{9} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (3 a^2 b+\frac {a^3}{x^4}+3 a b^2 x^4+b^3 x^8\right ) \, dx \\ & = -\frac {a^3}{3 x^3}+3 a^2 b x+\frac {3}{5} a b^2 x^5+\frac {b^3 x^9}{9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=-\frac {a^3}{3 x^3}+3 a^2 b x+\frac {3}{5} a b^2 x^5+\frac {b^3 x^9}{9} \]
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Time = 3.80 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {a^{3}}{3 x^{3}}+3 a^{2} b x +\frac {3 a \,b^{2} x^{5}}{5}+\frac {b^{3} x^{9}}{9}\) | \(34\) |
risch | \(-\frac {a^{3}}{3 x^{3}}+3 a^{2} b x +\frac {3 a \,b^{2} x^{5}}{5}+\frac {b^{3} x^{9}}{9}\) | \(34\) |
norman | \(\frac {\frac {1}{9} b^{3} x^{12}+\frac {3}{5} a \,b^{2} x^{8}+3 a^{2} b \,x^{4}-\frac {1}{3} a^{3}}{x^{3}}\) | \(37\) |
gosper | \(-\frac {-5 b^{3} x^{12}-27 a \,b^{2} x^{8}-135 a^{2} b \,x^{4}+15 a^{3}}{45 x^{3}}\) | \(38\) |
parallelrisch | \(\frac {5 b^{3} x^{12}+27 a \,b^{2} x^{8}+135 a^{2} b \,x^{4}-15 a^{3}}{45 x^{3}}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=\frac {5 \, b^{3} x^{12} + 27 \, a b^{2} x^{8} + 135 \, a^{2} b x^{4} - 15 \, a^{3}}{45 \, x^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=- \frac {a^{3}}{3 x^{3}} + 3 a^{2} b x + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{9}}{9} \]
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none
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=\frac {1}{9} \, b^{3} x^{9} + \frac {3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac {a^{3}}{3 \, x^{3}} \]
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none
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=\frac {1}{9} \, b^{3} x^{9} + \frac {3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac {a^{3}}{3 \, x^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^4\right )^3}{x^4} \, dx=\frac {b^3\,x^9}{9}-\frac {a^3}{3\,x^3}+\frac {3\,a\,b^2\,x^5}{5}+3\,a^2\,b\,x \]
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