Integrand size = 24, antiderivative size = 50 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=-\frac {b (b d-a e) x}{e^2}+\frac {(a+b x)^2}{2 e}+\frac {(b d-a e)^2 \log (d+e x)}{e^3} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {(b d-a e)^2 \log (d+e x)}{e^3}-\frac {b x (b d-a e)}{e^2}+\frac {(a+b x)^2}{2 e} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^2}{d+e x} \, dx \\ & = \int \left (-\frac {b (b d-a e)}{e^2}+\frac {b (a+b x)}{e}+\frac {(-b d+a e)^2}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {b (b d-a e) x}{e^2}+\frac {(a+b x)^2}{2 e}+\frac {(b d-a e)^2 \log (d+e x)}{e^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {b e x (-2 b d+4 a e+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \]
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Time = 2.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {b \left (\frac {1}{2} b e \,x^{2}+2 a e x -b d x \right )}{e^{2}}+\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(56\) |
norman | \(\frac {b \left (2 a e -b d \right ) x}{e^{2}}+\frac {b^{2} x^{2}}{2 e}+\frac {\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(59\) |
parallelrisch | \(\frac {x^{2} b^{2} e^{2}+2 \ln \left (e x +d \right ) a^{2} e^{2}-4 \ln \left (e x +d \right ) a b d e +2 \ln \left (e x +d \right ) b^{2} d^{2}+4 x a b \,e^{2}-2 b^{2} d e x}{2 e^{3}}\) | \(73\) |
risch | \(\frac {b^{2} x^{2}}{2 e}+\frac {2 b a x}{e}-\frac {b^{2} d x}{e^{2}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}-\frac {2 \ln \left (e x +d \right ) a b d}{e^{2}}+\frac {\ln \left (e x +d \right ) b^{2} d^{2}}{e^{3}}\) | \(74\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.24 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {b^{2} e^{2} x^{2} - 2 \, {\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
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Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {b^{2} x^{2}}{2 e} + x \left (\frac {2 a b}{e} - \frac {b^{2} d}{e^{2}}\right ) + \frac {\left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.20 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {b^{2} e x^{2} - 2 \, {\left (b^{2} d - 2 \, a b e\right )} x}{2 \, e^{2}} + \frac {{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.20 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {b^{2} e x^{2} - 2 \, b^{2} d x + 4 \, a b e x}{2 \, e^{2}} + \frac {{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{3}} \]
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Time = 10.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.24 \[ \int \frac {a^2+2 a b x+b^2 x^2}{d+e x} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{e^3}-x\,\left (\frac {b^2\,d}{e^2}-\frac {2\,a\,b}{e}\right )+\frac {b^2\,x^2}{2\,e} \]
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