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

Optimal. Leaf size=188 \[ \frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)} \]

[Out]

(e*f + d*g)^2/(32*d^4*e^3*(d - e*x)^2) + (f*(e*f + d*g))/(8*d^5*e^2*(d - e*x)) -
(e*f - d*g)^2/(24*d^3*e^3*(d + e*x)^3) - ((e*f - d*g)*(3*e*f + d*g))/(32*d^4*e^
3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)) + ((5*e^2*f^2 + 2*
d*e*f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(16*d^6*e^3)

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Rubi [A]  time = 0.457558, antiderivative size = 188, 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 \[ \frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(e*f + d*g)^2/(32*d^4*e^3*(d - e*x)^2) + (f*(e*f + d*g))/(8*d^5*e^2*(d - e*x)) -
(e*f - d*g)^2/(24*d^3*e^3*(d + e*x)^3) - ((e*f - d*g)*(3*e*f + d*g))/(32*d^4*e^
3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)) + ((5*e^2*f^2 + 2*
d*e*f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(16*d^6*e^3)

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Rubi in Sympy [A]  time = 68.3837, size = 206, normalized size = 1.1 \[ - \frac{\left (d g - e f\right )^{2}}{24 d^{3} e^{3} \left (d + e x\right )^{3}} + \frac{\left (d g - e f\right ) \left (d g + 3 e f\right )}{32 d^{4} e^{3} \left (d + e x\right )^{2}} + \frac{\left (d g + e f\right )^{2}}{32 d^{4} e^{3} \left (d - e x\right )^{2}} + \frac{f \left (d g + e f\right )}{8 d^{5} e^{2} \left (d - e x\right )} + \frac{d^{2} g^{2} - 3 e^{2} f^{2}}{16 d^{5} e^{3} \left (d + e x\right )} + \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (d - e x \right )}}{32 d^{6} e^{3}} - \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (d + e x \right )}}{32 d^{6} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d*g - e*f)**2/(24*d**3*e**3*(d + e*x)**3) + (d*g - e*f)*(d*g + 3*e*f)/(32*d**4
*e**3*(d + e*x)**2) + (d*g + e*f)**2/(32*d**4*e**3*(d - e*x)**2) + f*(d*g + e*f)
/(8*d**5*e**2*(d - e*x)) + (d**2*g**2 - 3*e**2*f**2)/(16*d**5*e**3*(d + e*x)) +
(d**2*g**2 - 2*d*e*f*g - 5*e**2*f**2)*log(d - e*x)/(32*d**6*e**3) - (d**2*g**2 -
2*d*e*f*g - 5*e**2*f**2)*log(d + e*x)/(32*d**6*e**3)

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Mathematica [A]  time = 0.29076, size = 197, normalized size = 1.05 \[ \frac{-\frac{4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac{3 d^2 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}+\frac{6 d \left (d^2 g^2-3 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-2 d e f g-5 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \log (d+e x)+\frac{3 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{12 d e f (d g+e f)}{d-e x}}{96 d^6 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((3*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (12*d*e*f*(e*f + d*g))/(d - e*x) - (4*d^3*(
e*f - d*g)^2)/(d + e*x)^3 + (3*d^2*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2))/(d + e*x)
^2 + (6*d*(-3*e^2*f^2 + d^2*g^2))/(d + e*x) + 3*(-5*e^2*f^2 - 2*d*e*f*g + d^2*g^
2)*Log[d - e*x] + 3*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(96*d^6*e^3)

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Maple [A]  time = 0.021, size = 348, normalized size = 1.9 \[{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{32\,e{d}^{4} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}{d}^{3} \left ( ex+d \right ) }}-{\frac{3\,{f}^{2}}{16\,e{d}^{5} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{{g}^{2}}{24\,d{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{12\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{24\,e{d}^{3} \left ( ex+d \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32/e^3/d^2/(e*x-d)^2*g^2+1/16/e^2/d^3/(e*x-d)^2*f*g+1/32/e/d^4/(e*x-d)^2*f^2+1
/32/e^3/d^4*ln(e*x-d)*g^2-1/16/e^2/d^5*ln(e*x-d)*f*g-5/32/e/d^6*ln(e*x-d)*f^2-1/
8/e^2/d^4/(e*x-d)*f*g-1/8/e/d^5/(e*x-d)*f^2+1/16/e^3/d^3/(e*x+d)*g^2-3/16/e*f^2/
d^5/(e*x+d)+1/32/e^3/d^2/(e*x+d)^2*g^2+1/16/e^2/d^3/(e*x+d)^2*f*g-3/32/e/d^4/(e*
x+d)^2*f^2-1/32/e^3/d^4*ln(e*x+d)*g^2+1/16/e^2/d^5*ln(e*x+d)*f*g+5/32/e/d^6*ln(e
*x+d)*f^2-1/24/e^3/d/(e*x+d)^3*g^2+1/12/e^2/d^2/(e*x+d)^3*f*g-1/24/e/d^3/(e*x+d)
^3*f^2

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Maxima [A]  time = 0.717695, size = 416, normalized size = 2.21 \[ -\frac{8 \, d^{4} e^{2} f^{2} - 16 \, d^{5} e f g - 4 \, d^{6} g^{2} + 3 \,{\left (5 \, e^{6} f^{2} + 2 \, d e^{5} f g - d^{2} e^{4} g^{2}\right )} x^{4} + 3 \,{\left (5 \, d e^{5} f^{2} + 2 \, d^{2} e^{4} f g - d^{3} e^{3} g^{2}\right )} x^{3} - 5 \,{\left (5 \, d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g - d^{4} e^{2} g^{2}\right )} x^{2} -{\left (25 \, d^{3} e^{3} f^{2} + 10 \, d^{4} e^{2} f g + 7 \, d^{5} e g^{2}\right )} x}{48 \,{\left (d^{5} e^{8} x^{5} + d^{6} e^{7} x^{4} - 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} + d^{9} e^{4} x + d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{6} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(8*d^4*e^2*f^2 - 16*d^5*e*f*g - 4*d^6*g^2 + 3*(5*e^6*f^2 + 2*d*e^5*f*g - d
^2*e^4*g^2)*x^4 + 3*(5*d*e^5*f^2 + 2*d^2*e^4*f*g - d^3*e^3*g^2)*x^3 - 5*(5*d^2*e
^4*f^2 + 2*d^3*e^3*f*g - d^4*e^2*g^2)*x^2 - (25*d^3*e^3*f^2 + 10*d^4*e^2*f*g + 7
*d^5*e*g^2)*x)/(d^5*e^8*x^5 + d^6*e^7*x^4 - 2*d^7*e^6*x^3 - 2*d^8*e^5*x^2 + d^9*
e^4*x + d^10*e^3) + 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(e*x + d)/(d^6*e^3
) - 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(e*x - d)/(d^6*e^3)

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Fricas [A]  time = 0.284789, size = 894, normalized size = 4.76 \[ -\frac{16 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 8 \, d^{7} g^{2} + 6 \,{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 6 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \,{\left (d^{6} e^{8} x^{5} + d^{7} e^{7} x^{4} - 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} + d^{10} e^{4} x + d^{11} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*(16*d^5*e^2*f^2 - 32*d^6*e*f*g - 8*d^7*g^2 + 6*(5*d*e^6*f^2 + 2*d^2*e^5*f*
g - d^3*e^4*g^2)*x^4 + 6*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 10*
(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 - 2*(25*d^4*e^3*f^2 + 10*d^5*e
^2*f*g + 7*d^6*e*g^2)*x - 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2
+ 2*d*e^6*f*g - d^2*e^5*g^2)*x^5 + (5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x
^4 - 2*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 2*(5*d^3*e^4*f^2 + 2*
d^4*e^3*f*g - d^5*e^2*g^2)*x^2 + (5*d^4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*e*g^2)*x)*
log(e*x + d) + 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2 + 2*d*e^6*f
*g - d^2*e^5*g^2)*x^5 + (5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(5*d
^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 2*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g
- d^5*e^2*g^2)*x^2 + (5*d^4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*e*g^2)*x)*log(e*x - d
))/(d^6*e^8*x^5 + d^7*e^7*x^4 - 2*d^8*e^6*x^3 - 2*d^9*e^5*x^2 + d^10*e^4*x + d^1
1*e^3)

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Sympy [A]  time = 7.60997, size = 320, normalized size = 1.7 \[ \frac{4 d^{6} g^{2} + 16 d^{5} e f g - 8 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} - 6 d e^{5} f g - 15 e^{6} f^{2}\right ) + x^{3} \left (3 d^{3} e^{3} g^{2} - 6 d^{2} e^{4} f g - 15 d e^{5} f^{2}\right ) + x^{2} \left (- 5 d^{4} e^{2} g^{2} + 10 d^{3} e^{3} f g + 25 d^{2} e^{4} f^{2}\right ) + x \left (7 d^{5} e g^{2} + 10 d^{4} e^{2} f g + 25 d^{3} e^{3} f^{2}\right )}{48 d^{10} e^{3} + 48 d^{9} e^{4} x - 96 d^{8} e^{5} x^{2} - 96 d^{7} e^{6} x^{3} + 48 d^{6} e^{7} x^{4} + 48 d^{5} e^{8} x^{5}} + \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (- \frac{d}{e} + x \right )}}{32 d^{6} e^{3}} - \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (\frac{d}{e} + x \right )}}{32 d^{6} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*d**6*g**2 + 16*d**5*e*f*g - 8*d**4*e**2*f**2 + x**4*(3*d**2*e**4*g**2 - 6*d*e
**5*f*g - 15*e**6*f**2) + x**3*(3*d**3*e**3*g**2 - 6*d**2*e**4*f*g - 15*d*e**5*f
**2) + x**2*(-5*d**4*e**2*g**2 + 10*d**3*e**3*f*g + 25*d**2*e**4*f**2) + x*(7*d*
*5*e*g**2 + 10*d**4*e**2*f*g + 25*d**3*e**3*f**2))/(48*d**10*e**3 + 48*d**9*e**4
*x - 96*d**8*e**5*x**2 - 96*d**7*e**6*x**3 + 48*d**6*e**7*x**4 + 48*d**5*e**8*x*
*5) + (d**2*g**2 - 2*d*e*f*g - 5*e**2*f**2)*log(-d/e + x)/(32*d**6*e**3) - (d**2
*g**2 - 2*d*e*f*g - 5*e**2*f**2)*log(d/e + x)/(32*d**6*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)^3*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError