Integrand size = 35, antiderivative size = 63 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2 d^2 x}{e^2}-\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3} \]
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Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=-\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac {c^2 d^2 x}{e^2} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x)^2}{(d+e x)^2} \, dx \\ & = \int \left (\frac {c^2 d^2}{e^2}+\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^2}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)}\right ) \, dx \\ & = \frac {c^2 d^2 x}{e^2}-\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2 d^2 e x-\frac {\left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (-c d^2+a e^2\right ) \log (d+e x)}{e^3} \]
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Time = 2.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {c^{2} d^{2} x}{e^{2}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{e^{3} \left (e x +d \right )}+\frac {2 c d \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(75\) |
risch | \(\frac {c^{2} d^{2} x}{e^{2}}-\frac {e \,a^{2}}{e x +d}+\frac {2 a c \,d^{2}}{e \left (e x +d \right )}-\frac {c^{2} d^{4}}{e^{3} \left (e x +d \right )}+\frac {2 c d \ln \left (e x +d \right ) a}{e}-\frac {2 c^{2} d^{3} \ln \left (e x +d \right )}{e^{3}}\) | \(92\) |
parallelrisch | \(\frac {2 \ln \left (e x +d \right ) x a c d \,e^{3}-2 \ln \left (e x +d \right ) x \,c^{2} d^{3} e +x^{2} c^{2} d^{2} e^{2}+2 \ln \left (e x +d \right ) a c \,d^{2} e^{2}-2 \ln \left (e x +d \right ) c^{2} d^{4}-a^{2} e^{4}+2 a c \,d^{2} e^{2}-2 c^{2} d^{4}}{e^{3} \left (e x +d \right )}\) | \(113\) |
norman | \(\frac {e \,c^{2} d^{2} x^{4}-\frac {d^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+4 c^{2} d^{4}\right )}{e^{3}}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) x^{2}}{e}-\frac {d \left (2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+10 c^{2} d^{4}\right ) x}{e^{2}}}{\left (e x +d \right )^{3}}+\frac {2 c d \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(149\) |
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Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} d^{2} e^{2} x^{2} + c^{2} d^{3} e x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 2 \, {\left (c^{2} d^{4} - a c d^{2} e^{2} + {\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} d^{2} x}{e^{2}} + \frac {2 c d \left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{4} + 2 a c d^{2} e^{2} - c^{2} d^{4}}{d e^{3} + e^{4} x} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} d^{2} x}{e^{2}} - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{4} x + d e^{3}} - \frac {2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} d^{2} x}{e^{2}} - \frac {2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{3}} - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{{\left (e x + d\right )} e^{3}} \]
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Time = 10.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2\,d^2\,x}{e^2}-\frac {a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}{e\,\left (x\,e^3+d\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{e^3} \]
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