Optimal. Leaf size=50 \[ \frac {g^2 x}{e^2}+\frac {(e f+d g)^2}{e^3 (d-e x)}+\frac {2 g (e f+d g) \log (d-e x)}{e^3} \]
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Rubi [A]
time = 0.04, 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 = {862, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 862
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 = 0.03, size = 46, normalized size = 0.92 \begin {gather*} \frac {e g^2 x+\frac {(e f+d g)^2}{d-e x}+2 g (e f+d g) \log (d-e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 63, normalized size = 1.26
method | result | size |
default | \(\frac {g^{2} x}{e^{2}}+\frac {2 g \left (d g +e f \right ) \ln \left (-e x +d \right )}{e^{3}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{e^{3} \left (-e x +d \right )}\) | \(63\) |
risch | \(\frac {g^{2} x}{e^{2}}+\frac {2 g^{2} \ln \left (-e x +d \right ) d}{e^{3}}+\frac {2 g \ln \left (-e x +d \right ) f}{e^{2}}+\frac {d^{2} g^{2}}{e^{3} \left (-e x +d \right )}+\frac {2 d f g}{e^{2} \left (-e x +d \right )}+\frac {f^{2}}{e \left (-e x +d \right )}\) | \(89\) |
norman | \(\frac {\frac {d \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right )}{e^{3}}+\frac {\left (2 d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) x}{e^{2}}-g^{2} x^{3}}{-e^{2} x^{2}+d^{2}}+\frac {2 g \left (d g +e f \right ) \ln \left (-e x +d \right )}{e^{3}}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 67, normalized size = 1.34 \begin {gather*} g^{2} x e^{\left (-2\right )} + 2 \, {\left (d g^{2} + f g e\right )} e^{\left (-3\right )} \log \left (x e - d\right ) - \frac {d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}}{x e^{4} - d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.92, size = 96, normalized size = 1.92 \begin {gather*} -\frac {d^{2} g^{2} - {\left (g^{2} x^{2} - f^{2}\right )} e^{2} + {\left (d g^{2} x + 2 \, d f g\right )} e + 2 \, {\left (d^{2} g^{2} - f g x e^{2} - {\left (d g^{2} x - d f g\right )} e\right )} \log \left (x e - d\right )}{x e^{4} - d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 61, normalized size = 1.22 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 68, normalized size = 1.36 \begin {gather*} g^{2} x e^{\left (-2\right )} + 2 \, {\left (d g^{2} + f g e\right )} e^{\left (-3\right )} \log \left ({\left | x e - d \right |}\right ) - \frac {{\left (d^{2} g^{2} + 2 \, d f g e + f^{2} e^{2}\right )} e^{\left (-3\right )}}{x e - d} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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