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

Optimal. Leaf size=87 \[ \frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}-\frac{(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2} \]

[Out]

-(e*f - d*g)^2/(4*d*e^3*(d + e*x)^2) - ((e*f - d*g)*(e*f + 3*d*g))/(4*d^2*e^3*(d
+ e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(4*d^3*e^3)

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Rubi [A]  time = 0.21826, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}-\frac{(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}-\frac{(e f-d g)^2}{4 d e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(e*f - d*g)^2/(4*d*e^3*(d + e*x)^2) - ((e*f - d*g)*(e*f + 3*d*g))/(4*d^2*e^3*(d
+ e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(4*d^3*e^3)

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Rubi in Sympy [A]  time = 43.5138, size = 73, normalized size = 0.84 \[ - \frac{\left (d g - e f\right )^{2}}{4 d e^{3} \left (d + e x\right )^{2}} + \frac{\left (d g - e f\right ) \left (3 d g + e f\right )}{4 d^{2} e^{3} \left (d + e x\right )} + \frac{\left (d g + e f\right )^{2} \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{4 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.127547, size = 87, normalized size = 1. \[ \frac{\frac{2 d (d g-e f) \left (2 d^2 g+d e (2 f+3 g x)+e^2 f x\right )}{(d+e x)^2}+(d g+e f)^2 (-\log (d-e x))+(d g+e f)^2 \log (d+e x)}{8 d^3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*d*(-(e*f) + d*g)*(2*d^2*g + e^2*f*x + d*e*(2*f + 3*g*x)))/(d + e*x)^2 - (e*f
+ d*g)^2*Log[d - e*x] + (e*f + d*g)^2*Log[d + e*x])/(8*d^3*e^3)

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Maple [B]  time = 0.015, size = 206, normalized size = 2.4 \[ -{\frac{\ln \left ( ex-d \right ){g}^{2}}{8\,d{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{4\,{d}^{2}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{8\,{d}^{3}e}}+{\frac{3\,{g}^{2}}{4\,{e}^{3} \left ( ex+d \right ) }}-{\frac{fg}{2\,{e}^{2}d \left ( ex+d \right ) }}-{\frac{{f}^{2}}{4\,{d}^{2}e \left ( ex+d \right ) }}-{\frac{{g}^{2}d}{4\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{fg}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{4\,de \left ( ex+d \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{8\,d{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{4\,{d}^{2}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{8\,{d}^{3}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/e^3/d*ln(e*x-d)*g^2-1/4/e^2/d^2*ln(e*x-d)*f*g-1/8/e/d^3*ln(e*x-d)*f^2+3/4/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^2-1/4/e^3*d/(e*x+d)^2*g
^2+1/2/e^2/(e*x+d)^2*f*g-1/4/e/d/(e*x+d)^2*f^2+1/8/e^3/d*ln(e*x+d)*g^2+1/4/e^2/d
^2*ln(e*x+d)*f*g+1/8/e/d^3*ln(e*x+d)*f^2

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Maxima [A]  time = 0.692041, size = 201, normalized size = 2.31 \[ -\frac{2 \, d e^{2} f^{2} - 2 \, d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x}{4 \,{\left (d^{2} e^{5} x^{2} + 2 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{8 \, d^{3} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{8 \, d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*d*e^2*f^2 - 2*d^3*g^2 + (e^3*f^2 + 2*d*e^2*f*g - 3*d^2*e*g^2)*x)/(d^2*e^
5*x^2 + 2*d^3*e^4*x + d^4*e^3) + 1/8*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x + d
)/(d^3*e^3) - 1/8*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^3*e^3)

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Fricas [A]  time = 0.280624, size = 366, normalized size = 4.21 \[ -\frac{4 \, d^{2} e^{2} f^{2} - 4 \, d^{4} g^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g - 3 \, d^{3} e g^{2}\right )} x -{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{8 \,{\left (d^{3} e^{5} x^{2} + 2 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*d^2*e^2*f^2 - 4*d^4*g^2 + 2*(d*e^3*f^2 + 2*d^2*e^2*f*g - 3*d^3*e*g^2)*x
- (d^2*e^2*f^2 + 2*d^3*e*f*g + d^4*g^2 + (e^4*f^2 + 2*d*e^3*f*g + d^2*e^2*g^2)*x
^2 + 2*(d*e^3*f^2 + 2*d^2*e^2*f*g + d^3*e*g^2)*x)*log(e*x + d) + (d^2*e^2*f^2 +
2*d^3*e*f*g + d^4*g^2 + (e^4*f^2 + 2*d*e^3*f*g + d^2*e^2*g^2)*x^2 + 2*(d*e^3*f^2
+ 2*d^2*e^2*f*g + d^3*e*g^2)*x)*log(e*x - d))/(d^3*e^5*x^2 + 2*d^4*e^4*x + d^5*
e^3)

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Sympy [A]  time = 4.06054, size = 185, normalized size = 2.13 \[ \frac{2 d^{3} g^{2} - 2 d e^{2} f^{2} + x \left (3 d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}\right )}{4 d^{4} e^{3} + 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac{\left (d g + e f\right )^{2} \log{\left (- \frac{d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac{\left (d g + e f\right )^{2} \log{\left (\frac{d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*d**3*g**2 - 2*d*e**2*f**2 + x*(3*d**2*e*g**2 - 2*d*e**2*f*g - e**3*f**2))/(4*
d**4*e**3 + 8*d**3*e**4*x + 4*d**2*e**5*x**2) - (d*g + e*f)**2*log(-d*(d*g + e*f
)**2/(e*(d**2*g**2 + 2*d*e*f*g + e**2*f**2)) + x)/(8*d**3*e**3) + (d*g + e*f)**2
*log(d*(d*g + e*f)**2/(e*(d**2*g**2 + 2*d*e*f*g + e**2*f**2)) + x)/(8*d**3*e**3)

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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)^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError