3.552 \(\int \frac{(f+g x)^2}{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{(d g+e f)^2 \log (d-e x)}{2 d e^3}+\frac{(e f-d g)^2 \log (d+e x)}{2 d e^3}-\frac{g^2 x}{e^2} \]

[Out]

-((g^2*x)/e^2) - ((e*f + d*g)^2*Log[d - e*x])/(2*d*e^3) + ((e*f - d*g)^2*Log[d +
e*x])/(2*d*e^3)

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Rubi [A]  time = 0.158385, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{(d g+e f)^2 \log (d-e x)}{2 d e^3}+\frac{(e f-d g)^2 \log (d+e x)}{2 d e^3}-\frac{g^2 x}{e^2} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)^2/(d^2 - e^2*x^2),x]

[Out]

-((g^2*x)/e^2) - ((e*f + d*g)^2*Log[d - e*x])/(2*d*e^3) + ((e*f - d*g)^2*Log[d +
e*x])/(2*d*e^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - g^{2} \int \frac{1}{e^{2}}\, dx + \frac{\left (d g - e f\right )^{2} \log{\left (d + e x \right )}}{2 d e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{2 d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**2/(-e**2*x**2+d**2),x)

[Out]

-g**2*Integral(e**(-2), x) + (d*g - e*f)**2*log(d + e*x)/(2*d*e**3) - (d*g + e*f
)**2*log(d - e*x)/(2*d*e**3)

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Mathematica [A]  time = 0.0381196, size = 55, normalized size = 0.89 \[ \frac{\left (d^2 g^2+e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )-d e g \left (f \log \left (d^2-e^2 x^2\right )+g x\right )}{d e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)^2/(d^2 - e^2*x^2),x]

[Out]

((e^2*f^2 + d^2*g^2)*ArcTanh[(e*x)/d] - d*e*g*(g*x + f*Log[d^2 - e^2*x^2]))/(d*e
^3)

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Maple [A]  time = 0.01, size = 107, normalized size = 1.7 \[ -{\frac{{g}^{2}x}{{e}^{2}}}-{\frac{d\ln \left ( ex-d \right ){g}^{2}}{2\,{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{2\,de}}+{\frac{d\ln \left ( ex+d \right ){g}^{2}}{2\,{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) fg}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{2\,de}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^2/(-e^2*x^2+d^2),x)

[Out]

-g^2*x/e^2-1/2/e^3*d*ln(e*x-d)*g^2-1/e^2*ln(e*x-d)*f*g-1/2/e/d*ln(e*x-d)*f^2+1/2
/e^3*d*ln(e*x+d)*g^2-1/e^2*ln(e*x+d)*f*g+1/2/e/d*ln(e*x+d)*f^2

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Maxima [A]  time = 0.689586, size = 111, normalized size = 1.79 \[ -\frac{g^{2} x}{e^{2}} + \frac{{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{2 \, d e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="maxima")

[Out]

-g^2*x/e^2 + 1/2*(e^2*f^2 - 2*d*e*f*g + d^2*g^2)*log(e*x + d)/(d*e^3) - 1/2*(e^2
*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x - d)/(d*e^3)

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Fricas [A]  time = 0.277806, size = 103, normalized size = 1.66 \[ -\frac{2 \, d e g^{2} x -{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right ) +{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="fricas")

[Out]

-1/2*(2*d*e*g^2*x - (e^2*f^2 - 2*d*e*f*g + d^2*g^2)*log(e*x + d) + (e^2*f^2 + 2*
d*e*f*g + d^2*g^2)*log(e*x - d))/(d*e^3)

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Sympy [A]  time = 2.68079, size = 112, normalized size = 1.81 \[ - \frac{g^{2} x}{e^{2}} + \frac{\left (d g - e f\right )^{2} \log{\left (x + \frac{2 d^{2} f g + \frac{d \left (d g - e f\right )^{2}}{e}}{d^{2} g^{2} + e^{2} f^{2}} \right )}}{2 d e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (x + \frac{2 d^{2} f g - \frac{d \left (d g + e f\right )^{2}}{e}}{d^{2} g^{2} + e^{2} f^{2}} \right )}}{2 d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**2/(-e**2*x**2+d**2),x)

[Out]

-g**2*x/e**2 + (d*g - e*f)**2*log(x + (2*d**2*f*g + d*(d*g - e*f)**2/e)/(d**2*g*
*2 + e**2*f**2))/(2*d*e**3) - (d*g + e*f)**2*log(x + (2*d**2*f*g - d*(d*g + e*f)
**2/e)/(d**2*g**2 + e**2*f**2))/(2*d*e**3)

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GIAC/XCAS [A]  time = 0.27282, size = 109, normalized size = 1.76 \[ -g^{2} x e^{\left (-2\right )} - f g e^{\left (-2\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{{\left (d^{2} g^{2} + f^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(g*x + f)^2/(e^2*x^2 - d^2),x, algorithm="giac")

[Out]

-g^2*x*e^(-2) - f*g*e^(-2)*ln(abs(x^2*e^2 - d^2)) - 1/2*(d^2*g^2 + f^2*e^2)*e^(-
3)*ln(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d)