Optimal. Leaf size=50 \[ -\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{g x (d g+e f)}{e^2}-\frac{(f+g x)^2}{2 e} \]
[Out]
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Rubi [A] time = 0.0676834, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{g x (d g+e f)}{e^2}-\frac{(f+g x)^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(f + g*x)^2)/(d^2 - e^2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\left (f + g x\right )^{2}}{2 e} - \frac{\left (d g + e f\right ) \int g\, dx}{e^{2}} - \frac{\left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(g*x+f)**2/(-e**2*x**2+d**2),x)
[Out]
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Mathematica [A] time = 0.032685, size = 43, normalized size = 0.86 \[ -\frac{e g x (2 d g+4 e f+e g x)+2 (d g+e f)^2 \log (d-e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(f + g*x)^2)/(d^2 - e^2*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 82, normalized size = 1.6 \[ -{\frac{{g}^{2}{x}^{2}}{2\,e}}-{\frac{{g}^{2}dx}{{e}^{2}}}-2\,{\frac{gfx}{e}}-{\frac{\ln \left ( ex-d \right ){d}^{2}{g}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex-d \right ) dfg}{{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(g*x+f)^2/(-e^2*x^2+d^2),x)
[Out]
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Maxima [A] time = 0.691315, size = 85, normalized size = 1.7 \[ -\frac{e g^{2} x^{2} + 2 \,{\left (2 \, e f g + d g^{2}\right )} x}{2 \, e^{2}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273369, size = 86, normalized size = 1.72 \[ -\frac{e^{2} g^{2} x^{2} + 2 \,{\left (2 \, e^{2} f g + d e g^{2}\right )} x + 2 \,{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.5712, size = 46, normalized size = 0.92 \[ - \frac{g^{2} x^{2}}{2 e} - \frac{x \left (d g^{2} + 2 e f g\right )}{e^{2}} - \frac{\left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(g*x+f)**2/(-e**2*x**2+d**2),x)
[Out]
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GIAC/XCAS [A] time = 0.275754, size = 181, normalized size = 3.62 \[ -\frac{1}{2} \,{\left (d^{2} g^{2} e + 2 \, d f g e^{2} + f^{2} e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{2} \,{\left (g^{2} x^{2} e^{3} + 2 \, d g^{2} x e^{2} + 4 \, f g x e^{3}\right )} e^{\left (-4\right )} - \frac{{\left (d^{3} g^{2} + 2 \, d^{2} f g e + d f^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)*(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="giac")
[Out]