Optimal. Leaf size=86 \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]
[Out]
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Rubi [A] time = 0.203801, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 31.9988, size = 75, normalized size = 0.87 \[ - \frac{\left (d g - e f\right )^{2}}{2 d e^{3} \left (d + e x\right )} - \frac{\left (d g - e f\right ) \left (3 d g + e f\right ) \log{\left (d + e x \right )}}{4 d^{2} e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{4 d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)**2/(e*x+d)/(-e**2*x**2+d**2),x)
[Out]
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Mathematica [A] time = 0.080469, size = 82, normalized size = 0.95 \[ \frac{(e f-d g) ((d+e x) (3 d g+e f) \log (d+e x)+2 d (d g-e f))-(d+e x) (d g+e f)^2 \log (d-e x)}{4 d^2 e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)),x]
[Out]
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Maple [A] time = 0.017, size = 149, normalized size = 1.7 \[ -{\frac{\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{2\,{e}^{2}d}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{3\,\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{2\,{e}^{2}d}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{{g}^{2}d}{2\,{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{2\,de \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2),x)
[Out]
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Maxima [A] time = 0.69311, size = 153, normalized size = 1.78 \[ -\frac{e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}}{2 \,{\left (d e^{4} x + d^{2} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} + 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(g*x + f)^2/((e^2*x^2 - d^2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279178, size = 223, normalized size = 2.59 \[ -\frac{2 \, d e^{2} f^{2} - 4 \, d^{2} e f g + 2 \, d^{3} g^{2} -{\left (d e^{2} f^{2} + 2 \, d^{2} e f g - 3 \, d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \,{\left (d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(g*x + f)^2/((e^2*x^2 - d^2)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.10636, size = 182, normalized size = 2.12 \[ - \frac{d^{2} g^{2} - 2 d e f g + e^{2} f^{2}}{2 d^{2} e^{3} + 2 d e^{4} x} - \frac{\left (d g - e f\right ) \left (3 d g + e f\right ) \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g - e f\right ) \left (3 d g + e f\right )}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g + e f\right )^{2}}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)**2/(e*x+d)/(-e**2*x**2+d**2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(g*x + f)^2/((e^2*x^2 - d^2)*(e*x + d)),x, algorithm="giac")
[Out]