Integrand size = 9, antiderivative size = 30 \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^4}+\frac {1}{b (a+b x)^3}\right ) \, dx \\ & = \frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {a+3 b x}{6 b^2 (a+b x)^3} \]
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Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {3 b x +a}{6 \left (b x +a \right )^{3} b^{2}}\) | \(19\) |
norman | \(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(22\) |
risch | \(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(22\) |
parallelrisch | \(\frac {-3 b^{2} x -a b}{6 b^{3} \left (b x +a \right )^{3}}\) | \(24\) |
default | \(\frac {a}{3 b^{2} \left (b x +a \right )^{3}}-\frac {1}{2 b^{2} \left (b x +a \right )^{2}}\) | \(27\) |
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none
Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {- a - 3 b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]
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none
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b x + a\right )}^{3} b^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {\frac {a}{6\,b^2}+\frac {x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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