Integrand size = 11, antiderivative size = 46 \[ \int \frac {x^3}{(a+b x)^2} \, dx=-\frac {2 a x}{b^3}+\frac {x^2}{2 b^2}+\frac {a^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4} \]
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Time = 0.02 (sec) , antiderivative size = 46, 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} \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {a^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{2 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx \\ & = -\frac {2 a x}{b^3}+\frac {x^2}{2 b^2}+\frac {a^3}{b^4 (a+b x)}+\frac {3 a^2 \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {-4 a b x+b^2 x^2+\frac {2 a^3}{a+b x}+6 a^2 \log (a+b x)}{2 b^4} \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98
method | result | size |
risch | \(-\frac {2 a x}{b^{3}}+\frac {x^{2}}{2 b^{2}}+\frac {a^{3}}{b^{4} \left (b x +a \right )}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}\) | \(45\) |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+2 a x}{b^{3}}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {a^{3}}{b^{4} \left (b x +a \right )}\) | \(46\) |
norman | \(\frac {\frac {3 a^{3}}{b^{4}}+\frac {x^{3}}{2 b}-\frac {3 a \,x^{2}}{2 b^{2}}}{b x +a}+\frac {3 a^{2} \ln \left (b x +a \right )}{b^{4}}\) | \(50\) |
parallelrisch | \(\frac {b^{3} x^{3}+6 \ln \left (b x +a \right ) x \,a^{2} b -3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )+6 a^{3}}{2 b^{4} \left (b x +a \right )}\) | \(59\) |
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none
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \, {\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {a^{3}}{a b^{4} + b^{5} x} + \frac {3 a^{2} \log {\left (a + b x \right )}}{b^{4}} - \frac {2 a x}{b^{3}} + \frac {x^{2}}{2 b^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {a^{3}}{b^{5} x + a b^{4}} + \frac {3 \, a^{2} \log \left (b x + a\right )}{b^{4}} + \frac {b x^{2} - 4 \, a x}{2 \, b^{3}} \]
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none
Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43 \[ \int \frac {x^3}{(a+b x)^2} \, dx=-\frac {{\left (b x + a\right )}^{2} {\left (\frac {6 \, a}{b x + a} - 1\right )}}{2 \, b^{4}} - \frac {3 \, a^{2} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {a^{3}}{{\left (b x + a\right )} b^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{(a+b x)^2} \, dx=\frac {x^2}{2\,b^2}+\frac {3\,a^2\,\ln \left (a+b\,x\right )}{b^4}+\frac {a^3}{b\,\left (x\,b^4+a\,b^3\right )}-\frac {2\,a\,x}{b^3} \]
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