3.568 \(\int \frac{(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=210 \[ \frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4} \]

[Out]

(e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) -
(e^2*f^2 - d^2*g^2)/(16*d^3*e^3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d
^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(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)*(3*e*f + d*g)*ArcTanh[(e*x)/d
])/(32*d^7*e^3)

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Rubi [A]  time = 0.521798, antiderivative size = 210, 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 \[ \frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2),x]

[Out]

(e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) -
(e^2*f^2 - d^2*g^2)/(16*d^3*e^3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d
^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(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)*(3*e*f + d*g)*ArcTanh[(e*x)/d
])/(32*d^7*e^3)

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Rubi in Sympy [A]  time = 75.9566, size = 218, normalized size = 1.04 \[ - \frac{\left (d g - e f\right )^{2}}{20 d^{2} e^{3} \left (d + e x\right )^{5}} + \frac{\left (d g - e f\right ) \left (d g + e f\right )}{16 d^{3} e^{3} \left (d + e x\right )^{4}} + \frac{\left (d g - 3 e f\right ) \left (d g + e f\right )}{48 d^{4} e^{3} \left (d + e x\right )^{3}} - \frac{f \left (d g + e f\right )}{16 d^{5} e^{2} \left (d + e x\right )^{2}} - \frac{\left (d g + e f\right ) \left (d g + 5 e f\right )}{64 d^{6} e^{3} \left (d + e x\right )} + \frac{\left (d g + e f\right )^{2}}{64 d^{6} e^{3} \left (d - e x\right )} - \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (d - e x \right )}}{64 d^{7} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (d + e x \right )}}{64 d^{7} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**2/(e*x+d)**4/(-e**2*x**2+d**2)**2,x)

[Out]

-(d*g - e*f)**2/(20*d**2*e**3*(d + e*x)**5) + (d*g - e*f)*(d*g + e*f)/(16*d**3*e
**3*(d + e*x)**4) + (d*g - 3*e*f)*(d*g + e*f)/(48*d**4*e**3*(d + e*x)**3) - f*(d
*g + e*f)/(16*d**5*e**2*(d + e*x)**2) - (d*g + e*f)*(d*g + 5*e*f)/(64*d**6*e**3*
(d + e*x)) + (d*g + e*f)**2/(64*d**6*e**3*(d - e*x)) - (d*g + e*f)*(d*g + 3*e*f)
*log(d - e*x)/(64*d**7*e**3) + (d*g + e*f)*(d*g + 3*e*f)*log(d + e*x)/(64*d**7*e
**3)

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Mathematica [A]  time = 0.348466, size = 229, normalized size = 1.09 \[ \frac{-\frac{48 d^5 (e f-d g)^2}{(d+e x)^5}-\frac{15 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d+e x}-15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d-e x)+15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac{60 d^2 e f (d g+e f)}{(d+e x)^2}+\frac{60 d^4 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^4}+\frac{20 d^3 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}+\frac{15 d (d g+e f)^2}{d-e x}}{960 d^7 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2),x]

[Out]

((15*d*(e*f + d*g)^2)/(d - e*x) - (48*d^5*(e*f - d*g)^2)/(d + e*x)^5 + (60*d^4*(
-(e^2*f^2) + d^2*g^2))/(d + e*x)^4 + (20*d^3*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))
/(d + e*x)^3 - (60*d^2*e*f*(e*f + d*g))/(d + e*x)^2 - (15*d*(5*e^2*f^2 + 6*d*e*f
*g + d^2*g^2))/(d + e*x) - 15*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d - e*x] + 1
5*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d + e*x])/(960*d^7*e^3)

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Maple [B]  time = 0.026, size = 394, normalized size = 1.9 \[ -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{6}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{32\,{e}^{2}{d}^{5} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{64\,e{d}^{6} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{6}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex+d \right ) }}-{\frac{3\,fg}{32\,{e}^{2}{d}^{5} \left ( ex+d \right ) }}-{\frac{5\,{f}^{2}}{64\,e{d}^{6} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}d \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{g}^{2}}{48\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{fg}{24\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{16\,e{d}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{g}^{2}}{20\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{fg}{10\,d{e}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{{f}^{2}}{20\,e{d}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{16\,e{d}^{5} \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x)

[Out]

-1/64/e^3/d^5*ln(e*x-d)*g^2-1/16/e^2/d^6*ln(e*x-d)*f*g-3/64/e/d^7*ln(e*x-d)*f^2-
1/64/e^3/d^4/(e*x-d)*g^2-1/32/e^2/d^5/(e*x-d)*f*g-1/64/e/d^6/(e*x-d)*f^2+1/64/e^
3/d^5*ln(e*x+d)*g^2+1/16/e^2/d^6*ln(e*x+d)*f*g+3/64/e/d^7*ln(e*x+d)*f^2-1/64/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+1/16/e^3/d/(e*x
+d)^4*g^2-1/16/e/d^3/(e*x+d)^4*f^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^2-1/20/e^3/(e*x+d)^5*g^2+1/10/e^2/d/(e*x+d)^5*f*
g-1/20/e/d^2/(e*x+d)^5*f^2-1/16/e^2*f/d^4/(e*x+d)^2*g-1/16/e*f^2/d^5/(e*x+d)^2

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Maxima [A]  time = 0.718927, size = 462, normalized size = 2.2 \[ \frac{144 \, d^{5} e^{2} f^{2} + 32 \, d^{6} e f g - 16 \, d^{7} g^{2} - 15 \,{\left (3 \, e^{7} f^{2} + 4 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 60 \,{\left (3 \, d e^{6} f^{2} + 4 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 80 \,{\left (3 \, d^{2} e^{5} f^{2} + 4 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 20 \,{\left (3 \, d^{3} e^{4} f^{2} + 4 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} +{\left (141 \, d^{4} e^{3} f^{2} + 188 \, d^{5} e^{2} f g - 49 \, d^{6} e g^{2}\right )} x}{480 \,{\left (d^{6} e^{9} x^{6} + 4 \, d^{7} e^{8} x^{5} + 5 \, d^{8} e^{7} x^{4} - 5 \, d^{10} e^{5} x^{2} - 4 \, d^{11} e^{4} x - d^{12} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{7} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{7} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^2/((e^2*x^2 - d^2)^2*(e*x + d)^4),x, algorithm="maxima")

[Out]

1/480*(144*d^5*e^2*f^2 + 32*d^6*e*f*g - 16*d^7*g^2 - 15*(3*e^7*f^2 + 4*d*e^6*f*g
+ d^2*e^5*g^2)*x^5 - 60*(3*d*e^6*f^2 + 4*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 80*(3
*d^2*e^5*f^2 + 4*d^3*e^4*f*g + d^4*e^3*g^2)*x^3 - 20*(3*d^3*e^4*f^2 + 4*d^4*e^3*
f*g + d^5*e^2*g^2)*x^2 + (141*d^4*e^3*f^2 + 188*d^5*e^2*f*g - 49*d^6*e*g^2)*x)/(
d^6*e^9*x^6 + 4*d^7*e^8*x^5 + 5*d^8*e^7*x^4 - 5*d^10*e^5*x^2 - 4*d^11*e^4*x - d^
12*e^3) + 1/64*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*log(e*x + d)/(d^7*e^3) - 1/64*(
3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^7*e^3)

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Fricas [A]  time = 0.283642, size = 936, normalized size = 4.46 \[ \frac{288 \, d^{6} e^{2} f^{2} + 64 \, d^{7} e f g - 32 \, d^{8} g^{2} - 30 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 120 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} - 160 \,{\left (3 \, d^{3} e^{5} f^{2} + 4 \, d^{4} e^{4} f g + d^{5} e^{3} g^{2}\right )} x^{3} - 40 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (141 \, d^{5} e^{3} f^{2} + 188 \, d^{6} e^{2} f g - 49 \, d^{7} e g^{2}\right )} x - 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{960 \,{\left (d^{7} e^{9} x^{6} + 4 \, d^{8} e^{8} x^{5} + 5 \, d^{9} e^{7} x^{4} - 5 \, d^{11} e^{5} x^{2} - 4 \, d^{12} e^{4} x - d^{13} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^2/((e^2*x^2 - d^2)^2*(e*x + d)^4),x, algorithm="fricas")

[Out]

1/960*(288*d^6*e^2*f^2 + 64*d^7*e*f*g - 32*d^8*g^2 - 30*(3*d*e^7*f^2 + 4*d^2*e^6
*f*g + d^3*e^5*g^2)*x^5 - 120*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4
- 160*(3*d^3*e^5*f^2 + 4*d^4*e^4*f*g + d^5*e^3*g^2)*x^3 - 40*(3*d^4*e^4*f^2 + 4*
d^5*e^3*f*g + d^6*e^2*g^2)*x^2 + 2*(141*d^5*e^3*f^2 + 188*d^6*e^2*f*g - 49*d^7*e
*g^2)*x - 15*(3*d^6*e^2*f^2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g +
d^2*e^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 5*(3*d^2
*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g +
d^6*e^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x + d
) + 15*(3*d^6*e^2*f^2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g + d^2*e
^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 5*(3*d^2*e^6*f
^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e
^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x - d))/(d^
7*e^9*x^6 + 4*d^8*e^8*x^5 + 5*d^9*e^7*x^4 - 5*d^11*e^5*x^2 - 4*d^12*e^4*x - d^13
*e^3)

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Sympy [A]  time = 8.98622, size = 420, normalized size = 2. \[ - \frac{16 d^{7} g^{2} - 32 d^{6} e f g - 144 d^{5} e^{2} f^{2} + x^{5} \left (15 d^{2} e^{5} g^{2} + 60 d e^{6} f g + 45 e^{7} f^{2}\right ) + x^{4} \left (60 d^{3} e^{4} g^{2} + 240 d^{2} e^{5} f g + 180 d e^{6} f^{2}\right ) + x^{3} \left (80 d^{4} e^{3} g^{2} + 320 d^{3} e^{4} f g + 240 d^{2} e^{5} f^{2}\right ) + x^{2} \left (20 d^{5} e^{2} g^{2} + 80 d^{4} e^{3} f g + 60 d^{3} e^{4} f^{2}\right ) + x \left (49 d^{6} e g^{2} - 188 d^{5} e^{2} f g - 141 d^{4} e^{3} f^{2}\right )}{- 480 d^{12} e^{3} - 1920 d^{11} e^{4} x - 2400 d^{10} e^{5} x^{2} + 2400 d^{8} e^{7} x^{4} + 1920 d^{7} e^{8} x^{5} + 480 d^{6} e^{9} x^{6}} - \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**2/(e*x+d)**4/(-e**2*x**2+d**2)**2,x)

[Out]

-(16*d**7*g**2 - 32*d**6*e*f*g - 144*d**5*e**2*f**2 + x**5*(15*d**2*e**5*g**2 +
60*d*e**6*f*g + 45*e**7*f**2) + x**4*(60*d**3*e**4*g**2 + 240*d**2*e**5*f*g + 18
0*d*e**6*f**2) + x**3*(80*d**4*e**3*g**2 + 320*d**3*e**4*f*g + 240*d**2*e**5*f**
2) + x**2*(20*d**5*e**2*g**2 + 80*d**4*e**3*f*g + 60*d**3*e**4*f**2) + x*(49*d**
6*e*g**2 - 188*d**5*e**2*f*g - 141*d**4*e**3*f**2))/(-480*d**12*e**3 - 1920*d**1
1*e**4*x - 2400*d**10*e**5*x**2 + 2400*d**8*e**7*x**4 + 1920*d**7*e**8*x**5 + 48
0*d**6*e**9*x**6) - (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 + 4*d*e*f*g + 3*e**2*f**2)) + x)/(64*d**7*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 + 4*d*e*f*g + 3*e**2*f**2
)) + x)/(64*d**7*e**3)

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GIAC/XCAS [A]  time = 0.28349, 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)^2*(e*x + d)^4),x, algorithm="giac")

[Out]

Done