Optimal. Leaf size=50 \[ \frac {(d g+e f)^2}{e^3 (d-e x)}+\frac {2 g (d g+e f) \log (d-e x)}{e^3}+\frac {g^2 x}{e^2} \]
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Rubi [A] time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {848, 43} \[ \frac {(d g+e f)^2}{e^3 (d-e x)}+\frac {2 g (d g+e f) \log (d-e x)}{e^3}+\frac {g^2 x}{e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {g^2}{e^2}+\frac {(e f+d g)^2}{e^2 (-d+e x)^2}+\frac {2 g (e f+d g)}{e^2 (-d+e x)}\right ) \, dx\\ &=\frac {g^2 x}{e^2}+\frac {(e f+d g)^2}{e^3 (d-e x)}+\frac {2 g (e f+d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 0.92 \[ \frac {\frac {(d g+e f)^2}{d-e x}+2 g (d g+e f) \log (d-e x)+e g^2 x}{e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 95, normalized size = 1.90 \[ \frac {e^{2} g^{2} x^{2} - d e g^{2} x - e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2} - 2 \, {\left (d e f g + d^{2} g^{2} - {\left (e^{2} f g + d e g^{2}\right )} x\right )} \log \left (e x - d\right )}{e^{4} x - d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 160, normalized size = 3.20 \[ g^{2} x e^{\left (-2\right )} + {\left (d g^{2} e + f g e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {{\left (d^{2} g^{2} e^{2} + d f g e^{3}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {{\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3} + {\left (d^{2} g^{2} e^{2} + 2 \, d f g e^{3} + f^{2} e^{4}\right )} x\right )} e^{\left (-4\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.92 \[ -\frac {d^{2} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {2 d f g}{\left (e x -d \right ) e^{2}}+\frac {2 d \,g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {f^{2}}{\left (e x -d \right ) e}+\frac {2 f g \ln \left (e x -d \right )}{e^{2}}+\frac {g^{2} x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 69, normalized size = 1.38 \[ \frac {g^{2} x}{e^{2}} - \frac {e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}}{e^{4} x - d e^{3}} + \frac {2 \, {\left (e f g + d g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.56, size = 72, normalized size = 1.44 \[ \frac {d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {g^2\,x}{e^2}+\frac {\ln \left (e\,x-d\right )\,\left (2\,d\,g^2+2\,e\,f\,g\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 61, normalized size = 1.22 \[ \frac {- d^{2} g^{2} - 2 d e f g - e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac {g^{2} x}{e^{2}} + \frac {2 g \left (d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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