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

Optimal. Leaf size=122 \[ \frac{(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)} \]

[Out]

(e*f + d*g)^2/(8*d^2*e^3*(d - e*x)^2) + (e^2*f^2 - d^2*g^2)/(4*d^3*e^3*(d - e*x)
) - (e*f - d*g)^2/(8*d^3*e^3*(d + e*x)) + ((e*f - d*g)*(3*e*f + d*g)*ArcTanh[(e*
x)/d])/(8*d^4*e^3)

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Rubi [A]  time = 0.263918, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(d g+3 e f) (e f-d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}-\frac{(e f-d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2}{8 d^2 e^3 (d-e x)^2}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d-e x)} \]

Antiderivative was successfully verified.

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

[Out]

(e*f + d*g)^2/(8*d^2*e^3*(d - e*x)^2) + (e^2*f^2 - d^2*g^2)/(4*d^3*e^3*(d - e*x)
) - (e*f - d*g)^2/(8*d^3*e^3*(d + e*x)) + ((e*f - d*g)*(3*e*f + d*g)*ArcTanh[(e*
x)/d])/(8*d^4*e^3)

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Rubi in Sympy [A]  time = 46.6515, size = 133, normalized size = 1.09 \[ \frac{\left (d g + e f\right )^{2}}{8 d^{2} e^{3} \left (d - e x\right )^{2}} - \frac{\left (d g - e f\right )^{2}}{8 d^{3} e^{3} \left (d + e x\right )} - \frac{\left (d g - e f\right ) \left (d g + e f\right )}{4 d^{3} e^{3} \left (d - e x\right )} + \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (d - e x \right )}}{16 d^{4} e^{3}} - \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (d + e x \right )}}{16 d^{4} e^{3}} \]

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)**3,x)

[Out]

(d*g + e*f)**2/(8*d**2*e**3*(d - e*x)**2) - (d*g - e*f)**2/(8*d**3*e**3*(d + e*x
)) - (d*g - e*f)*(d*g + e*f)/(4*d**3*e**3*(d - e*x)) + (d*g - e*f)*(d*g + 3*e*f)
*log(d - e*x)/(16*d**4*e**3) - (d*g - e*f)*(d*g + 3*e*f)*log(d + e*x)/(16*d**4*e
**3)

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Mathematica [A]  time = 0.212078, size = 140, normalized size = 1.15 \[ \frac{\frac{4 d e^2 f^2-4 d^3 g^2}{d-e x}+\left (d^2 g^2+2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2-2 d e f g+3 e^2 f^2\right ) \log (d+e x)+\frac{2 d^2 (d g+e f)^2}{(d-e x)^2}-\frac{2 d (e f-d g)^2}{d+e x}}{16 d^4 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (4*d*e^2*f^2 - 4*d^3*g^2)/(d - e*x) - (2*d*
(e*f - d*g)^2)/(d + e*x) + (-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + (3*
e^2*f^2 - 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(16*d^4*e^3)

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Maple [B]  time = 0.019, size = 257, normalized size = 2.1 \[{\frac{{g}^{2}}{4\,d{e}^{3} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{3} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{8\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{4\,d{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{8\,e{d}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}+{\frac{\ln \left ( ex-d \right ) fg}{8\,{e}^{2}{d}^{3}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) fg}{8\,{e}^{2}{d}^{3}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{{g}^{2}}{8\,d{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{4\,{e}^{2}{d}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{3} \left ( ex+d \right ) }} \]

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)^3,x)

[Out]

1/4/e^3/d/(e*x-d)*g^2-1/4/e/d^3/(e*x-d)*f^2+1/8/e^3/(e*x-d)^2*g^2+1/4/e^2/d/(e*x
-d)^2*f*g+1/8/e/d^2/(e*x-d)^2*f^2+1/16/e^3/d^2*ln(e*x-d)*g^2+1/8/e^2/d^3*ln(e*x-
d)*f*g-3/16/e/d^4*ln(e*x-d)*f^2-1/16/e^3/d^2*ln(e*x+d)*g^2-1/8/e^2/d^3*ln(e*x+d)
*f*g+3/16/e/d^4*ln(e*x+d)*f^2-1/8/d/e^3/(e*x+d)*g^2+1/4/d^2/e^2/(e*x+d)*f*g-1/8/
d^3/e/(e*x+d)*f^2

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Maxima [A]  time = 0.703857, size = 285, normalized size = 2.34 \[ \frac{2 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} -{\left (3 \, e^{4} f^{2} - 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} +{\left (3 \, d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \,{\left (d^{3} e^{6} x^{3} - d^{4} e^{5} x^{2} - d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \]

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)^3,x, algorithm="maxima")

[Out]

1/8*(2*d^2*e^2*f^2 + 4*d^3*e*f*g - 2*d^4*g^2 - (3*e^4*f^2 - 2*d*e^3*f*g - d^2*e^
2*g^2)*x^2 + (3*d*e^3*f^2 - 2*d^2*e^2*f*g + 3*d^3*e*g^2)*x)/(d^3*e^6*x^3 - d^4*e
^5*x^2 - d^5*e^4*x + d^6*e^3) + 1/16*(3*e^2*f^2 - 2*d*e*f*g - d^2*g^2)*log(e*x +
d)/(d^4*e^3) - 1/16*(3*e^2*f^2 - 2*d*e*f*g - d^2*g^2)*log(e*x - d)/(d^4*e^3)

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Fricas [A]  time = 0.278086, size = 563, normalized size = 4.61 \[ \frac{4 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \,{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x +{\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} +{\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} -{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) -{\left (3 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - d^{5} g^{2} +{\left (3 \, e^{5} f^{2} - 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} - 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} -{\left (3 \, d^{2} e^{3} f^{2} - 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{16 \,{\left (d^{4} e^{6} x^{3} - d^{5} e^{5} x^{2} - d^{6} e^{4} x + d^{7} e^{3}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/16*(4*d^3*e^2*f^2 + 8*d^4*e*f*g - 4*d^5*g^2 - 2*(3*d*e^4*f^2 - 2*d^2*e^3*f*g -
d^3*e^2*g^2)*x^2 + 2*(3*d^2*e^3*f^2 - 2*d^3*e^2*f*g + 3*d^4*e*g^2)*x + (3*d^3*e
^2*f^2 - 2*d^4*e*f*g - d^5*g^2 + (3*e^5*f^2 - 2*d*e^4*f*g - d^2*e^3*g^2)*x^3 - (
3*d*e^4*f^2 - 2*d^2*e^3*f*g - d^3*e^2*g^2)*x^2 - (3*d^2*e^3*f^2 - 2*d^3*e^2*f*g
- d^4*e*g^2)*x)*log(e*x + d) - (3*d^3*e^2*f^2 - 2*d^4*e*f*g - d^5*g^2 + (3*e^5*f
^2 - 2*d*e^4*f*g - d^2*e^3*g^2)*x^3 - (3*d*e^4*f^2 - 2*d^2*e^3*f*g - d^3*e^2*g^2
)*x^2 - (3*d^2*e^3*f^2 - 2*d^3*e^2*f*g - d^4*e*g^2)*x)*log(e*x - d))/(d^4*e^6*x^
3 - d^5*e^5*x^2 - d^6*e^4*x + d^7*e^3)

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Sympy [A]  time = 5.28868, size = 277, normalized size = 2.27 \[ \frac{- 2 d^{4} g^{2} + 4 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x^{2} \left (d^{2} e^{2} g^{2} + 2 d e^{3} f g - 3 e^{4} f^{2}\right ) + x \left (3 d^{3} e g^{2} - 2 d^{2} e^{2} f g + 3 d e^{3} f^{2}\right )}{8 d^{6} e^{3} - 8 d^{5} e^{4} x - 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (- \frac{d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac{\left (d g - e f\right ) \left (d g + 3 e f\right ) \log{\left (\frac{d \left (d g - e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} \]

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)**3,x)

[Out]

(-2*d**4*g**2 + 4*d**3*e*f*g + 2*d**2*e**2*f**2 + x**2*(d**2*e**2*g**2 + 2*d*e**
3*f*g - 3*e**4*f**2) + x*(3*d**3*e*g**2 - 2*d**2*e**2*f*g + 3*d*e**3*f**2))/(8*d
**6*e**3 - 8*d**5*e**4*x - 8*d**4*e**5*x**2 + 8*d**3*e**6*x**3) + (d*g - e*f)*(d
*g + 3*e*f)*log(-d*(d*g - e*f)*(d*g + 3*e*f)/(e*(d**2*g**2 + 2*d*e*f*g - 3*e**2*
f**2)) + x)/(16*d**4*e**3) - (d*g - e*f)*(d*g + 3*e*f)*log(d*(d*g - e*f)*(d*g +
3*e*f)/(e*(d**2*g**2 + 2*d*e*f*g - 3*e**2*f**2)) + x)/(16*d**4*e**3)

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GIAC/XCAS [A]  time = 0.270168, size = 258, normalized size = 2.11 \[ \frac{{\left (d^{2} g^{2} + 2 \, d f g e - 3 \, f^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{16 \, d^{3}{\left | d \right |}} + \frac{{\left (d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} + d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 2 \, d^{3} f g x e^{3} + 4 \, d^{4} f g e^{2} - 3 \, f^{2} x^{3} e^{6} + 5 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{3}} \]

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)^3,x, algorithm="giac")

[Out]

1/16*(d^2*g^2 + 2*d*f*g*e - 3*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^3*abs(d)) + 1/8*(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 + 2*d^3*f*g*x*e^3 + 4*d^4*f*g*e^2
- 3*f^2*x^3*e^6 + 5*d^2*f^2*x*e^4 + 2*d^3*f^2*e^3)*e^(-4)/((x^2*e^2 - d^2)^2*d^3
)