Integrand size = 31, antiderivative size = 59 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}-\frac {2 e (b d-a e)}{b^3 (a+b x)}+\frac {e^2 \log (a+b x)}{b^3} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 45} \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {2 e (b d-a e)}{b^3 (a+b x)}-\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}+\frac {e^2 \log (a+b x)}{b^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^2}{(a+b x)^3} \, dx \\ & = \int \left (\frac {(b d-a e)^2}{b^2 (a+b x)^3}+\frac {2 e (b d-a e)}{b^2 (a+b x)^2}+\frac {e^2}{b^2 (a+b x)}\right ) \, dx \\ & = -\frac {(b d-a e)^2}{2 b^3 (a+b x)^2}-\frac {2 e (b d-a e)}{b^3 (a+b x)}+\frac {e^2 \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-\frac {(b d-a e) (3 a e+b (d+4 e x))}{(a+b x)^2}+2 e^2 \log (a+b x)}{2 b^3} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {\frac {2 e \left (a e -b d \right ) x}{b^{2}}+\frac {3 e^{2} a^{2}-2 a b d e -b^{2} d^{2}}{2 b^{3}}}{\left (b x +a \right )^{2}}+\frac {e^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(67\) |
default | \(\frac {e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {e^{2} a^{2}-2 a b d e +b^{2} d^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {2 e \left (a e -b d \right )}{b^{3} \left (b x +a \right )}\) | \(69\) |
norman | \(\frac {\frac {\left (2 e^{2} a -2 b d e \right ) x^{2}}{b}+\frac {a \left (3 e^{2} a^{2}-2 a b d e -b^{2} d^{2}\right )}{2 b^{3}}+\frac {\left (7 e^{2} a^{2}-6 a b d e -b^{2} d^{2}\right ) x}{2 b^{2}}}{\left (b x +a \right )^{3}}+\frac {e^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(101\) |
parallelrisch | \(\frac {2 \ln \left (b x +a \right ) x^{2} b^{2} e^{2}+4 \ln \left (b x +a \right ) x a b \,e^{2}+2 \ln \left (b x +a \right ) a^{2} e^{2}+4 a b \,e^{2} x -4 b^{2} d e x +3 e^{2} a^{2}-2 a b d e -b^{2} d^{2}}{2 b^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )}\) | \(108\) |
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Time = 0.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {3 a^{2} e^{2} - 2 a b d e - b^{2} d^{2} + x \left (4 a b e^{2} - 4 b^{2} d e\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {e^{2} \log {\left (a + b x \right )}}{b^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {e^{2} \log \left (b x + a\right )}{b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {4 \, {\left (b d e - a e^{2}\right )} x + \frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2}}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^2\,\ln \left (a+b\,x\right )}{b^3}-\frac {\frac {-3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2}{2\,b^3}-\frac {2\,e\,x\,\left (a\,e-b\,d\right )}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
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