Integrand size = 11, antiderivative size = 50 \[ \int \frac {x^3}{(a+b x)^3} \, dx=\frac {x}{b^3}+\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4} \]
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Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^3}{(a+b x)^3} \, dx=\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4}+\frac {x}{b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {x}{b^3}+\frac {a^3}{2 b^4 (a+b x)^2}-\frac {3 a^2}{b^4 (a+b x)}-\frac {3 a \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \frac {x^3}{(a+b x)^3} \, dx=-\frac {-2 b x+\frac {a^2 (5 a+6 b x)}{(a+b x)^2}+6 a \log (a+b x)}{2 b^4} \]
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Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {x}{b^{3}}+\frac {-3 a^{2} x -\frac {5 a^{3}}{2 b}}{b^{3} \left (b x +a \right )^{2}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(45\) |
norman | \(\frac {\frac {x^{3}}{b}-\frac {9 a^{3}}{2 b^{4}}-\frac {6 a^{2} x}{b^{3}}}{\left (b x +a \right )^{2}}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(47\) |
default | \(\frac {x}{b^{3}}+\frac {a^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {3 a^{2}}{b^{4} \left (b x +a \right )}-\frac {3 a \ln \left (b x +a \right )}{b^{4}}\) | \(49\) |
parallelrisch | \(-\frac {6 \ln \left (b x +a \right ) x^{2} a \,b^{2}-2 b^{3} x^{3}+12 \ln \left (b x +a \right ) x \,a^{2} b +6 a^{3} \ln \left (b x +a \right )+12 a^{2} b x +9 a^{3}}{2 b^{4} \left (b x +a \right )^{2}}\) | \(73\) |
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none
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \frac {x^3}{(a+b x)^3} \, dx=\frac {2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.16 \[ \int \frac {x^3}{(a+b x)^3} \, dx=- \frac {3 a \log {\left (a + b x \right )}}{b^{4}} + \frac {- 5 a^{3} - 6 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {x}{b^{3}} \]
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none
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{(a+b x)^3} \, dx=-\frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {x}{b^{3}} - \frac {3 \, a \log \left (b x + a\right )}{b^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{(a+b x)^3} \, dx=\frac {x}{b^{3}} - \frac {3 \, a \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x + 5 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} b^{4}} \]
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Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{(a+b x)^3} \, dx=-\frac {3\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {3\,a^2}{a+b\,x}-\frac {a^3}{2\,{\left (a+b\,x\right )}^2}}{b^4} \]
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