3.574 \(\int \frac{(d+e x)^2 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=88 \[ \frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}+\frac{(e f-3 d g) (d g+e f)}{4 d^2 e^3 (d-e x)}+\frac{(d g+e f)^2}{4 d e^3 (d-e x)^2} \]

[Out]

(e*f + d*g)^2/(4*d*e^3*(d - e*x)^2) + ((e*f - 3*d*g)*(e*f + 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 = 0.213714, antiderivative size = 88, 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 \[ \frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}+\frac{(e f-3 d g) (d g+e f)}{4 d^2 e^3 (d-e x)}+\frac{(d g+e f)^2}{4 d e^3 (d-e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*f + d*g)^2/(4*d*e^3*(d - e*x)^2) + ((e*f - 3*d*g)*(e*f + 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 = 43.458, size = 73, normalized size = 0.83 \[ \frac{\left (d g + e f\right )^{2}}{4 d e^{3} \left (d - e x\right )^{2}} - \frac{\left (d g + e f\right ) \left (3 d g - e f\right )}{4 d^{2} e^{3} \left (d - e x\right )} + \frac{\left (d g - e f\right )^{2} \operatorname{atanh}{\left (\frac{e x}{d} \right )}}{4 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(g*x+f)**2/(-e**2*x**2+d**2)**3,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 = 0.133155, size = 90, normalized size = 1.02 \[ \frac{-\frac{2 d (d g+e f) \left (2 d^2 g-d e (2 f+3 g x)+e^2 f x\right )}{(d-e x)^2}+(e f-d g)^2 (-\log (d-e x))+(e f-d g)^2 \log (d+e x)}{8 d^3 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^2*(f + g*x)^2)/(d^2 - e^2*x^2)^3,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 = 218, normalized size = 2.5 \[{\frac{3\,{g}^{2}}{4\,{e}^{3} \left ( ex-d \right ) }}+{\frac{fg}{2\,{e}^{2}d \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,{d}^{2}e \left ( ex-d \right ) }}+{\frac{{g}^{2}d}{4\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{2\,{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{4\,de \left ( ex-d \right ) ^{2}}}-{\frac{\ln \left ( ex-d \right ){g}^{2}}{8\,d{e}^{3}}}+{\frac{\ln \left ( ex-d \right ) fg}{4\,{d}^{2}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{8\,{d}^{3}e}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{8\,d{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) fg}{4\,{d}^{2}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{8\,{d}^{3}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4/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/(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+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 = 0.709806, size = 203, 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(-(e*x + d)^2*(g*x + f)^2/(e^2*x^2 - d^2)^3,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 = 0.279323, size = 366, normalized size = 4.16 \[ \frac{4 \, d^{2} e^{2} f^{2} - 4 \, d^{4} g^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g - 3 \, d^{3} e g^{2}\right )} x +{\left (d^{2} e^{2} f^{2} - 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x + d\right ) -{\left (d^{2} e^{2} f^{2} - 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{8 \,{\left (d^{3} e^{5} x^{2} - 2 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^2*(g*x + f)^2/(e^2*x^2 - d^2)^3,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 = 4.24326, size = 185, normalized size = 2.1 \[ \frac{- 2 d^{3} g^{2} + 2 d e^{2} f^{2} + x \left (3 d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}\right )}{4 d^{4} e^{3} - 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac{\left (d g - e f\right )^{2} \log{\left (- \frac{d \left (d g - e f\right )^{2}}{e \left (d^{2} g^{2} - 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac{\left (d g - e f\right )^{2} \log{\left (\frac{d \left (d g - e f\right )^{2}}{e \left (d^{2} g^{2} - 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(g*x+f)**2/(-e**2*x**2+d**2)**3,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 [A]  time = 0.276341, size = 266, normalized size = 3.02 \[ -\frac{{\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{8 \, d^{2}{\left | d \right |}} + \frac{{\left (3 \, d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} - d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 4 \, d^{2} f g x^{2} e^{4} + 2 \, d^{3} f g x e^{3} - f^{2} x^{3} e^{6} + 3 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{4 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^2*(g*x + f)^2/(e^2*x^2 - d^2)^3,x, algorithm="giac")

[Out]

-1/8*(d^2*g^2*e^2 - 2*d*f*g*e^3 + f^2*e^4)*e^(-5)*ln(abs(2*x*e^2 - 2*abs(d)*e)/a
bs(2*x*e^2 + 2*abs(d)*e))/(d^2*abs(d)) + 1/4*(3*d^2*g^2*x^3*e^4 + 4*d^3*g^2*x^2*
e^3 - d^4*g^2*x*e^2 - 2*d^5*g^2*e + 2*d*f*g*x^3*e^5 + 4*d^2*f*g*x^2*e^4 + 2*d^3*
f*g*x*e^3 - f^2*x^3*e^6 + 3*d^2*f^2*x*e^4 + 2*d^3*f^2*e^3)*e^(-4)/((x^2*e^2 - d^
2)^2*d^2)