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

Optimal. Leaf size=74 \[ \frac{(e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3}+\frac{(f+g x) \left (d^2 g+e^2 f x\right )}{2 d^2 e^2 \left (d^2-e^2 x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0735644, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3}+\frac{(f+g x) \left (d^2 g+e^2 f x\right )}{2 d^2 e^2 \left (d^2-e^2 x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.4728, size = 63, normalized size = 0.85 \[ \frac{\left (f + g x\right ) \left (d^{2} g + e^{2} f x\right )}{2 d^{2} e^{2} \left (d^{2} - e^{2} x^{2}\right )} - \frac{\left (d^{2} g^{2} - e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{2 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0632949, size = 85, normalized size = 1.15 \[ \frac{-2 d^2 f g-d^2 g^2 x-e^2 f^2 x}{2 d^2 e^2 \left (e^2 x^2-d^2\right )}-\frac{\left (d^2 g^2-e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 180, normalized size = 2.4 \[{\frac{\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}d}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,e{d}^{3}}}-{\frac{{g}^{2}}{4\,{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{2\,{e}^{2}d \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{2} \left ( ex-d \right ) }}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}d}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,e{d}^{3}}}-{\frac{{g}^{2}}{4\,{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{2\,{e}^{2}d \left ( ex+d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{2} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/e^3/d*ln(e*x-d)*g^2-1/4/e/d^3*ln(e*x-d)*f^2-1/4/e^3/(e*x-d)*g^2-1/2/e^2/d/(e
*x-d)*f*g-1/4/e/d^2/(e*x-d)*f^2-1/4/e^3/d*ln(e*x+d)*g^2+1/4/e/d^3*ln(e*x+d)*f^2-
1/4/e^3/(e*x+d)*g^2+1/2/d/e^2/(e*x+d)*f*g-1/4/d^2/e/(e*x+d)*f^2

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Maxima [A]  time = 0.691804, size = 150, normalized size = 2.03 \[ -\frac{2 \, d^{2} f g +{\left (e^{2} f^{2} + d^{2} g^{2}\right )} x}{2 \,{\left (d^{2} e^{4} x^{2} - d^{4} e^{2}\right )}} + \frac{{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{3} e^{3}} - \frac{{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*d^2*f*g + (e^2*f^2 + d^2*g^2)*x)/(d^2*e^4*x^2 - d^4*e^2) + 1/4*(e^2*f^2
- d^2*g^2)*log(e*x + d)/(d^3*e^3) - 1/4*(e^2*f^2 - d^2*g^2)*log(e*x - d)/(d^3*e^
3)

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Fricas [A]  time = 0.278029, size = 209, normalized size = 2.82 \[ -\frac{4 \, d^{3} e f g + 2 \,{\left (d e^{3} f^{2} + d^{3} e g^{2}\right )} x +{\left (d^{2} e^{2} f^{2} - d^{4} g^{2} -{\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x + d\right ) -{\left (d^{2} e^{2} f^{2} - d^{4} g^{2} -{\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x - d\right )}{4 \,{\left (d^{3} e^{5} x^{2} - d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*d^3*e*f*g + 2*(d*e^3*f^2 + d^3*e*g^2)*x + (d^2*e^2*f^2 - d^4*g^2 - (e^4*
f^2 - d^2*e^2*g^2)*x^2)*log(e*x + d) - (d^2*e^2*f^2 - d^4*g^2 - (e^4*f^2 - d^2*e
^2*g^2)*x^2)*log(e*x - d))/(d^3*e^5*x^2 - d^5*e^3)

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Sympy [A]  time = 2.96306, size = 155, normalized size = 2.09 \[ - \frac{2 d^{2} f g + x \left (d^{2} g^{2} + e^{2} f^{2}\right )}{- 2 d^{4} e^{2} + 2 d^{2} e^{4} x^{2}} + \frac{\left (d g - e f\right ) \left (d g + e f\right ) \log{\left (- \frac{d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} - \frac{\left (d g - e f\right ) \left (d g + e f\right ) \log{\left (\frac{d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*d**2*f*g + x*(d**2*g**2 + e**2*f**2))/(-2*d**4*e**2 + 2*d**2*e**4*x**2) + (d
*g - e*f)*(d*g + e*f)*log(-d*(d*g - e*f)*(d*g + e*f)/(e*(d**2*g**2 - e**2*f**2))
+ x)/(4*d**3*e**3) - (d*g - e*f)*(d*g + e*f)*log(d*(d*g - e*f)*(d*g + e*f)/(e*(
d**2*g**2 - e**2*f**2)) + x)/(4*d**3*e**3)

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GIAC/XCAS [A]  time = 0.300546, size = 136, normalized size = 1.84 \[ \frac{{\left (d^{2} g^{2} - f^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{4 \, d^{2}{\left | d \right |}} - \frac{{\left (d^{2} g^{2} x + 2 \, d^{2} f g + f^{2} x e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e^{2} - d^{2}\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(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))/(d^2*abs(d)) - 1/2*(d^2*g^2*x + 2*d^2*f*g + f^2*x*e^2)*e^(-2)/((x^2*e^2 -
d^2)*d^2)