Integrand size = 17, antiderivative size = 33 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=\frac {x}{b^2}-\frac {a^2}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3} \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45} \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=-\frac {a^2}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3}+\frac {x}{b^2} \]
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Rule 45
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{(a+b x)^2} \, dx \\ & = \int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx \\ & = \frac {x}{b^2}-\frac {a^2}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=\frac {b x-\frac {a^2}{a+b x}-2 a \log (a+b x)}{b^3} \]
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Time = 1.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {x}{b^{2}}-\frac {a^{2}}{b^{3} \left (b x +a \right )}-\frac {2 a \ln \left (b x +a \right )}{b^{3}}\) | \(34\) |
risch | \(\frac {x}{b^{2}}-\frac {a^{2}}{b^{3} \left (b x +a \right )}-\frac {2 a \ln \left (b x +a \right )}{b^{3}}\) | \(34\) |
norman | \(\frac {\frac {x^{5}}{b}-\frac {2 a^{2} x^{3}}{b^{3}}}{x^{3} \left (b x +a \right )}-\frac {2 a \ln \left (b x +a \right )}{b^{3}}\) | \(44\) |
parallelrisch | \(-\frac {2 \ln \left (b x +a \right ) x a b -b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )+2 a^{2}}{b^{3} \left (b x +a \right )}\) | \(49\) |
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none
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=\frac {b^{2} x^{2} + a b x - a^{2} - 2 \, {\left (a b x + a^{2}\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=- \frac {a^{2}}{a b^{3} + b^{4} x} - \frac {2 a \log {\left (a + b x \right )}}{b^{3}} + \frac {x}{b^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=-\frac {a^{2}}{b^{4} x + a b^{3}} + \frac {x}{b^{2}} - \frac {2 \, a \log \left (b x + a\right )}{b^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=\frac {x}{b^{2}} - \frac {2 \, a \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {a^{2}}{{\left (b x + a\right )} b^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {x^6}{\left (a x^2+b x^3\right )^2} \, dx=\frac {x}{b^2}-\frac {a^2}{x\,b^4+a\,b^3}-\frac {2\,a\,\ln \left (a+b\,x\right )}{b^3} \]
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