Integrand size = 29, antiderivative size = 65 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {2 d g (e f+d g) x}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g}-\frac {2 d (e f+d g)^2 \log (d-e x)}{e^3} \]
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Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {862, 78} \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {2 d (d g+e f)^2 \log (d-e x)}{e^3}-\frac {2 d g x (d g+e f)}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g} \]
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Rule 78
Rule 862
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x) (f+g x)^2}{d-e x} \, dx \\ & = \int \left (-\frac {2 d g (e f+d g)}{e^2}-\frac {2 d (e f+d g)^2}{e^2 (-d+e x)}-\frac {2 d g (f+g x)}{e}-(f+g x)^2\right ) \, dx \\ & = -\frac {2 d g (e f+d g) x}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g}-\frac {2 d (e f+d g)^2 \log (d-e x)}{e^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {e x \left (6 d^2 g^2+3 d e g (4 f+g x)+e^2 \left (3 f^2+3 f g x+g^2 x^2\right )\right )+6 d (e f+d g)^2 \log (d-e x)}{3 e^3} \]
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Time = 0.42 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.35
method | result | size |
norman | \(-\frac {g^{2} x^{3}}{3}-\frac {\left (2 d^{2} g^{2}+4 d e f g +e^{2} f^{2}\right ) x}{e^{2}}-\frac {g \left (d g +e f \right ) x^{2}}{e}-\frac {2 d \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(88\) |
default | \(-\frac {\frac {1}{3} g^{2} x^{3} e^{2}+d e \,g^{2} x^{2}+e^{2} f g \,x^{2}+2 d^{2} g^{2} x +4 d e f g x +e^{2} f^{2} x}{e^{2}}-\frac {2 d \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(95\) |
risch | \(-\frac {g^{2} x^{3}}{3}-\frac {d \,g^{2} x^{2}}{e}-f g \,x^{2}-\frac {2 d^{2} g^{2} x}{e^{2}}-\frac {4 d f g x}{e}-f^{2} x -\frac {2 d^{3} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {4 d^{2} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {2 d \ln \left (-e x +d \right ) f^{2}}{e}\) | \(107\) |
parallelrisch | \(-\frac {g^{2} x^{3} e^{3}+3 x^{2} d \,e^{2} g^{2}+3 x^{2} e^{3} f g +6 \ln \left (e x -d \right ) d^{3} g^{2}+12 \ln \left (e x -d \right ) d^{2} e f g +6 \ln \left (e x -d \right ) d \,e^{2} f^{2}+6 x \,d^{2} e \,g^{2}+12 x d \,e^{2} f g +3 x \,e^{3} f^{2}}{3 e^{3}}\) | \(116\) |
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Time = 0.57 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {e^{3} g^{2} x^{3} + 3 \, {\left (e^{3} f g + d e^{2} g^{2}\right )} x^{2} + 3 \, {\left (e^{3} f^{2} + 4 \, d e^{2} f g + 2 \, d^{2} e g^{2}\right )} x + 6 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )} \log \left (e x - d\right )}{3 \, e^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=- \frac {2 d \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {g^{2} x^{3}}{3} - x^{2} \left (\frac {d g^{2}}{e} + f g\right ) - x \left (\frac {2 d^{2} g^{2}}{e^{2}} + \frac {4 d f g}{e} + f^{2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {e^{2} g^{2} x^{3} + 3 \, {\left (e^{2} f g + d e g^{2}\right )} x^{2} + 3 \, {\left (e^{2} f^{2} + 4 \, d e f g + 2 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} - \frac {2 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.62 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {2 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{e^{3}} - \frac {e^{3} g^{2} x^{3} + 3 \, e^{3} f g x^{2} + 3 \, d e^{2} g^{2} x^{2} + 3 \, e^{3} f^{2} x + 12 \, d e^{2} f g x + 6 \, d^{2} e g^{2} x}{3 \, e^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx=-x^2\,\left (\frac {d\,g^2+2\,e\,f\,g}{2\,e}+\frac {d\,g^2}{2\,e}\right )-x\,\left (\frac {e\,f^2+2\,d\,g\,f}{e}+\frac {d\,\left (\frac {d\,g^2+2\,e\,f\,g}{e}+\frac {d\,g^2}{e}\right )}{e}\right )-\frac {g^2\,x^3}{3}-\frac {\ln \left (e\,x-d\right )\,\left (2\,d^3\,g^2+4\,d^2\,e\,f\,g+2\,d\,e^2\,f^2\right )}{e^3} \]
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