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

Optimal. Leaf size=235 \[ \frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac{(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac{(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}+\frac{\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{64 d^7 e^3}-\frac{-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac{3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2} \]

[Out]

(e*f + d*g)^2/(64*d^5*e^3*(d - e*x)^2) + ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3
*(d - e*x)) - (e*f - d*g)^2/(32*d^3*e^3*(d + e*x)^4) - ((e*f - d*g)*(3*e*f + d*g
))/(48*d^4*e^3*(d + e*x)^3) - (3*e^2*f^2 - d^2*g^2)/(32*d^5*e^3*(d + e*x)^2) - (
5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)/(32*d^6*e^3*(d + e*x)) + ((15*e^2*f^2 + 10*d*e*
f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(64*d^7*e^3)

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Rubi [A]  time = 0.590926, antiderivative size = 235, 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 \[ \frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac{(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac{(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}+\frac{\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{64 d^7 e^3}-\frac{-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac{3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*f + d*g)^2/(64*d^5*e^3*(d - e*x)^2) + ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3
*(d - e*x)) - (e*f - d*g)^2/(32*d^3*e^3*(d + e*x)^4) - ((e*f - d*g)*(3*e*f + d*g
))/(48*d^4*e^3*(d + e*x)^3) - (3*e^2*f^2 - d^2*g^2)/(32*d^5*e^3*(d + e*x)^2) - (
5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)/(32*d^6*e^3*(d + e*x)) + ((15*e^2*f^2 + 10*d*e*
f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(64*d^7*e^3)

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Rubi in Sympy [A]  time = 86.4509, size = 252, normalized size = 1.07 \[ - \frac{\left (d g - e f\right )^{2}}{32 d^{3} e^{3} \left (d + e x\right )^{4}} + \frac{\left (d g - e f\right ) \left (d g + 3 e f\right )}{48 d^{4} e^{3} \left (d + e x\right )^{3}} + \frac{d^{2} g^{2} - 3 e^{2} f^{2}}{32 d^{5} e^{3} \left (d + e x\right )^{2}} + \frac{\left (d g + e f\right )^{2}}{64 d^{5} e^{3} \left (d - e x\right )^{2}} + \frac{d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}}{32 d^{6} e^{3} \left (d + e x\right )} + \frac{\left (d g + e f\right ) \left (d g + 5 e f\right )}{64 d^{6} e^{3} \left (d - e x\right )} + \frac{\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log{\left (d - e x \right )}}{128 d^{7} e^{3}} - \frac{\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log{\left (d + e x \right )}}{128 d^{7} e^{3}} \]

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)**3,x)

[Out]

-(d*g - e*f)**2/(32*d**3*e**3*(d + e*x)**4) + (d*g - e*f)*(d*g + 3*e*f)/(48*d**4
*e**3*(d + e*x)**3) + (d**2*g**2 - 3*e**2*f**2)/(32*d**5*e**3*(d + e*x)**2) + (d
*g + e*f)**2/(64*d**5*e**3*(d - e*x)**2) + (d**2*g**2 - 2*d*e*f*g - 5*e**2*f**2)
/(32*d**6*e**3*(d + e*x)) + (d*g + e*f)*(d*g + 5*e*f)/(64*d**6*e**3*(d - e*x)) +
(d**2*g**2 - 10*d*e*f*g - 15*e**2*f**2)*log(d - e*x)/(128*d**7*e**3) - (d**2*g*
*2 - 10*d*e*f*g - 15*e**2*f**2)*log(d + e*x)/(128*d**7*e**3)

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Mathematica [A]  time = 0.301928, size = 244, normalized size = 1.04 \[ \frac{-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{12 d^2 \left (d^2 g^2-3 e^2 f^2\right )}{(d+e x)^2}+\frac{6 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d-e x}+\frac{12 d \left (d^2 g^2-2 d e f g-5 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-10 d e f g-15 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \log (d+e x)+\frac{6 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{8 d^3 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}}{384 d^7 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((6*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (6*d*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d
- e*x) - (12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (8*d^3*(-3*e^2*f^2 + 2*d*e*f*g + d
^2*g^2))/(d + e*x)^3 + (12*d^2*(-3*e^2*f^2 + d^2*g^2))/(d + e*x)^2 + (12*d*(-5*e
^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x) + 3*(-15*e^2*f^2 - 10*d*e*f*g + d^2*g^2
)*Log[d - e*x] + 3*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*Log[d + e*x])/(384*d^7*e^
3)

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Maple [A]  time = 0.022, size = 421, normalized size = 1.8 \[ -{\frac{{g}^{2}}{64\,{e}^{3}{d}^{4} \left ( ex-d \right ) }}-{\frac{3\,fg}{32\,{e}^{2}{d}^{5} \left ( ex-d \right ) }}-{\frac{5\,{f}^{2}}{64\,e{d}^{6} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{64\,{d}^{3}{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{32\,{e}^{2}{d}^{4} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{64\,e{d}^{5} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{128\,{e}^{3}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ) fg}{64\,{e}^{2}{d}^{6}}}-{\frac{15\,\ln \left ( ex-d \right ){f}^{2}}{128\,e{d}^{7}}}+{\frac{{g}^{2}}{32\,{d}^{3}{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{g}^{2}}{48\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{24\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{16\,e{d}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{{g}^{2}}{32\,{e}^{3}{d}^{4} \left ( ex+d \right ) }}-{\frac{fg}{16\,{e}^{2}{d}^{5} \left ( ex+d \right ) }}-{\frac{5\,{f}^{2}}{32\,e{d}^{6} \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{128\,{e}^{3}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ) fg}{64\,{e}^{2}{d}^{6}}}+{\frac{15\,\ln \left ( ex+d \right ){f}^{2}}{128\,e{d}^{7}}}-{\frac{{g}^{2}}{32\,d{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{16\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{32\,e{d}^{3} \left ( ex+d \right ) ^{4}}} \]

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)^3,x)

[Out]

-1/64/e^3/d^4/(e*x-d)*g^2-3/32/e^2/d^5/(e*x-d)*f*g-5/64/e/d^6/(e*x-d)*f^2+1/64/e
^3/d^3/(e*x-d)^2*g^2+1/32/e^2/d^4/(e*x-d)^2*f*g+1/64/e/d^5/(e*x-d)^2*f^2+1/128/e
^3/d^5*ln(e*x-d)*g^2-5/64/e^2/d^6*ln(e*x-d)*f*g-15/128/e/d^7*ln(e*x-d)*f^2+1/32/
e^3/d^3/(e*x+d)^2*g^2-3/32/e*f^2/d^5/(e*x+d)^2+1/48/e^3/d^2/(e*x+d)^3*g^2+1/24/e
^2/d^3/(e*x+d)^3*f*g-1/16/e/d^4/(e*x+d)^3*f^2+1/32/e^3/d^4/(e*x+d)*g^2-1/16/e^2/
d^5/(e*x+d)*f*g-5/32/e/d^6/(e*x+d)*f^2-1/128/e^3/d^5*ln(e*x+d)*g^2+5/64/e^2/d^6*
ln(e*x+d)*f*g+15/128/e/d^7*ln(e*x+d)*f^2-1/32/e^3/d/(e*x+d)^4*g^2+1/16/e^2/d^2/(
e*x+d)^4*f*g-1/32/e/d^3/(e*x+d)^4*f^2

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Maxima [A]  time = 0.710225, size = 485, normalized size = 2.06 \[ -\frac{48 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 16 \, d^{7} g^{2} + 3 \,{\left (15 \, e^{7} f^{2} + 10 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + 6 \,{\left (15 \, d e^{6} f^{2} + 10 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (15 \, d^{2} e^{5} f^{2} + 10 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \,{\left (15 \, d^{3} e^{4} f^{2} + 10 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} -{\left (51 \, d^{4} e^{3} f^{2} + 34 \, d^{5} e^{2} f g + 35 \, d^{6} e g^{2}\right )} x}{192 \,{\left (d^{6} e^{9} x^{6} + 2 \, d^{7} e^{8} x^{5} - d^{8} e^{7} x^{4} - 4 \, d^{9} e^{6} x^{3} - d^{10} e^{5} x^{2} + 2 \, d^{11} e^{4} x + d^{12} e^{3}\right )}} + \frac{{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{128 \, d^{7} e^{3}} - \frac{{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{128 \, d^{7} e^{3}} \]

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)^2),x, algorithm="maxima")

[Out]

-1/192*(48*d^5*e^2*f^2 - 32*d^6*e*f*g - 16*d^7*g^2 + 3*(15*e^7*f^2 + 10*d*e^6*f*
g - d^2*e^5*g^2)*x^5 + 6*(15*d*e^6*f^2 + 10*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(
15*d^2*e^5*f^2 + 10*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 10*(15*d^3*e^4*f^2 + 10*d^4
*e^3*f*g - d^5*e^2*g^2)*x^2 - (51*d^4*e^3*f^2 + 34*d^5*e^2*f*g + 35*d^6*e*g^2)*x
)/(d^6*e^9*x^6 + 2*d^7*e^8*x^5 - d^8*e^7*x^4 - 4*d^9*e^6*x^3 - d^10*e^5*x^2 + 2*
d^11*e^4*x + d^12*e^3) + 1/128*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*log(e*x + d)/
(d^7*e^3) - 1/128*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*log(e*x - d)/(d^7*e^3)

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Fricas [A]  time = 0.286354, size = 1071, normalized size = 4.56 \[ \text{result too large to display} \]

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)^2),x, algorithm="fricas")

[Out]

-1/384*(96*d^6*e^2*f^2 - 64*d^7*e*f*g - 32*d^8*g^2 + 6*(15*d*e^7*f^2 + 10*d^2*e^
6*f*g - d^3*e^5*g^2)*x^5 + 12*(15*d^2*e^6*f^2 + 10*d^3*e^5*f*g - d^4*e^4*g^2)*x^
4 - 4*(15*d^3*e^5*f^2 + 10*d^4*e^4*f*g - d^5*e^3*g^2)*x^3 - 20*(15*d^4*e^4*f^2 +
10*d^5*e^3*f*g - d^6*e^2*g^2)*x^2 - 2*(51*d^5*e^3*f^2 + 34*d^6*e^2*f*g + 35*d^7
*e*g^2)*x - 3*(15*d^6*e^2*f^2 + 10*d^7*e*f*g - d^8*g^2 + (15*e^8*f^2 + 10*d*e^7*
f*g - d^2*e^6*g^2)*x^6 + 2*(15*d*e^7*f^2 + 10*d^2*e^6*f*g - d^3*e^5*g^2)*x^5 - (
15*d^2*e^6*f^2 + 10*d^3*e^5*f*g - d^4*e^4*g^2)*x^4 - 4*(15*d^3*e^5*f^2 + 10*d^4*
e^4*f*g - d^5*e^3*g^2)*x^3 - (15*d^4*e^4*f^2 + 10*d^5*e^3*f*g - d^6*e^2*g^2)*x^2
+ 2*(15*d^5*e^3*f^2 + 10*d^6*e^2*f*g - d^7*e*g^2)*x)*log(e*x + d) + 3*(15*d^6*e
^2*f^2 + 10*d^7*e*f*g - d^8*g^2 + (15*e^8*f^2 + 10*d*e^7*f*g - d^2*e^6*g^2)*x^6
+ 2*(15*d*e^7*f^2 + 10*d^2*e^6*f*g - d^3*e^5*g^2)*x^5 - (15*d^2*e^6*f^2 + 10*d^3
*e^5*f*g - d^4*e^4*g^2)*x^4 - 4*(15*d^3*e^5*f^2 + 10*d^4*e^4*f*g - d^5*e^3*g^2)*
x^3 - (15*d^4*e^4*f^2 + 10*d^5*e^3*f*g - d^6*e^2*g^2)*x^2 + 2*(15*d^5*e^3*f^2 +
10*d^6*e^2*f*g - d^7*e*g^2)*x)*log(e*x - d))/(d^7*e^9*x^6 + 2*d^8*e^8*x^5 - d^9*
e^7*x^4 - 4*d^10*e^6*x^3 - d^11*e^5*x^2 + 2*d^12*e^4*x + d^13*e^3)

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Sympy [A]  time = 8.92703, size = 371, normalized size = 1.58 \[ \frac{16 d^{7} g^{2} + 32 d^{6} e f g - 48 d^{5} e^{2} f^{2} + x^{5} \left (3 d^{2} e^{5} g^{2} - 30 d e^{6} f g - 45 e^{7} f^{2}\right ) + x^{4} \left (6 d^{3} e^{4} g^{2} - 60 d^{2} e^{5} f g - 90 d e^{6} f^{2}\right ) + x^{3} \left (- 2 d^{4} e^{3} g^{2} + 20 d^{3} e^{4} f g + 30 d^{2} e^{5} f^{2}\right ) + x^{2} \left (- 10 d^{5} e^{2} g^{2} + 100 d^{4} e^{3} f g + 150 d^{3} e^{4} f^{2}\right ) + x \left (35 d^{6} e g^{2} + 34 d^{5} e^{2} f g + 51 d^{4} e^{3} f^{2}\right )}{192 d^{12} e^{3} + 384 d^{11} e^{4} x - 192 d^{10} e^{5} x^{2} - 768 d^{9} e^{6} x^{3} - 192 d^{8} e^{7} x^{4} + 384 d^{7} e^{8} x^{5} + 192 d^{6} e^{9} x^{6}} + \frac{\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log{\left (- \frac{d}{e} + x \right )}}{128 d^{7} e^{3}} - \frac{\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log{\left (\frac{d}{e} + x \right )}}{128 d^{7} e^{3}} \]

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)**3,x)

[Out]

(16*d**7*g**2 + 32*d**6*e*f*g - 48*d**5*e**2*f**2 + x**5*(3*d**2*e**5*g**2 - 30*
d*e**6*f*g - 45*e**7*f**2) + x**4*(6*d**3*e**4*g**2 - 60*d**2*e**5*f*g - 90*d*e*
*6*f**2) + x**3*(-2*d**4*e**3*g**2 + 20*d**3*e**4*f*g + 30*d**2*e**5*f**2) + x**
2*(-10*d**5*e**2*g**2 + 100*d**4*e**3*f*g + 150*d**3*e**4*f**2) + x*(35*d**6*e*g
**2 + 34*d**5*e**2*f*g + 51*d**4*e**3*f**2))/(192*d**12*e**3 + 384*d**11*e**4*x
- 192*d**10*e**5*x**2 - 768*d**9*e**6*x**3 - 192*d**8*e**7*x**4 + 384*d**7*e**8*
x**5 + 192*d**6*e**9*x**6) + (d**2*g**2 - 10*d*e*f*g - 15*e**2*f**2)*log(-d/e +
x)/(128*d**7*e**3) - (d**2*g**2 - 10*d*e*f*g - 15*e**2*f**2)*log(d/e + x)/(128*d
**7*e**3)

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GIAC/XCAS [A]  time = 0.34375, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^2),x, algorithm="giac")

[Out]

Done