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

Optimal. Leaf size=107 \[ \frac{4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.291335, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 d^{2} \left (d g + e f\right )^{2}}{e^{3} \left (d - e x\right )} + \frac{4 d \left (d g + e f\right ) \left (3 d g + e f\right ) \log{\left (d - e x \right )}}{e^{3}} + \frac{g^{2} x^{3}}{3} + \frac{2 g \left (2 d g + e f\right ) \int x\, dx}{e} + \frac{\left (8 d g \left (d g + e f\right ) + e^{2} f^{2}\right ) \int f^{2}\, dx}{e^{2} f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.157385, size = 115, normalized size = 1.07 \[ -\frac{4 d^2 (d g+e f)^2}{e^3 (e x-d)}+\frac{x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac{4 d \left (3 d^2 g^2+4 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{g x^2 (2 d g+e f)}{e}+\frac{g^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 167, normalized size = 1.6 \[{\frac{{g}^{2}{x}^{3}}{3}}+2\,{\frac{d{x}^{2}{g}^{2}}{e}}+{x}^{2}fg+8\,{\frac{{d}^{2}{g}^{2}x}{{e}^{2}}}+8\,{\frac{dfgx}{e}}+{f}^{2}x+12\,{\frac{{d}^{3}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+16\,{\frac{{d}^{2}\ln \left ( ex-d \right ) fg}{{e}^{2}}}+4\,{\frac{d\ln \left ( ex-d \right ){f}^{2}}{e}}-4\,{\frac{{d}^{4}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-8\,{\frac{{d}^{3}fg}{{e}^{2} \left ( ex-d \right ) }}-4\,{\frac{{d}^{2}{f}^{2}}{e \left ( ex-d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*g^2*x^3+2/e*x^2*d*g^2+x^2*f*g+8/e^2*d^2*g^2*x+8/e*d*f*g*x+f^2*x+12*d^3/e^3*l
n(e*x-d)*g^2+16*d^2/e^2*ln(e*x-d)*f*g+4*d/e*ln(e*x-d)*f^2-4*d^4/e^3/(e*x-d)*g^2-
8*d^3/e^2/(e*x-d)*f*g-4*d^2/e/(e*x-d)*f^2

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Maxima [A]  time = 0.690636, size = 190, normalized size = 1.78 \[ -\frac{4 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{e^{2} g^{2} x^{3} + 3 \,{\left (e^{2} f g + 2 \, d e g^{2}\right )} x^{2} + 3 \,{\left (e^{2} f^{2} + 8 \, d e f g + 8 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} + \frac{4 \,{\left (d e^{2} f^{2} + 4 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27675, size = 278, normalized size = 2.6 \[ \frac{e^{4} g^{2} x^{4} - 12 \, d^{2} e^{2} f^{2} - 24 \, d^{3} e f g - 12 \, d^{4} g^{2} +{\left (3 \, e^{4} f g + 5 \, d e^{3} g^{2}\right )} x^{3} + 3 \,{\left (e^{4} f^{2} + 7 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{2} - 3 \,{\left (d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 8 \, d^{3} e g^{2}\right )} x - 12 \,{\left (d^{2} e^{2} f^{2} + 4 \, d^{3} e f g + 3 \, d^{4} g^{2} -{\left (d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \,{\left (e^{4} x - d e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(e^4*g^2*x^4 - 12*d^2*e^2*f^2 - 24*d^3*e*f*g - 12*d^4*g^2 + (3*e^4*f*g + 5*d
*e^3*g^2)*x^3 + 3*(e^4*f^2 + 7*d*e^3*f*g + 6*d^2*e^2*g^2)*x^2 - 3*(d*e^3*f^2 + 8
*d^2*e^2*f*g + 8*d^3*e*g^2)*x - 12*(d^2*e^2*f^2 + 4*d^3*e*f*g + 3*d^4*g^2 - (d*e
^3*f^2 + 4*d^2*e^2*f*g + 3*d^3*e*g^2)*x)*log(e*x - d))/(e^4*x - d*e^3)

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Sympy [A]  time = 3.14097, size = 122, normalized size = 1.14 \[ \frac{4 d \left (d g + e f\right ) \left (3 d g + e f\right ) \log{\left (- d + e x \right )}}{e^{3}} + \frac{g^{2} x^{3}}{3} - \frac{4 d^{4} g^{2} + 8 d^{3} e f g + 4 d^{2} e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{x^{2} \left (2 d g^{2} + e f g\right )}{e} + \frac{x \left (8 d^{2} g^{2} + 8 d e f g + e^{2} f^{2}\right )}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.281117, size = 338, normalized size = 3.16 \[ 2 \,{\left (3 \, d^{3} g^{2} e^{3} + 4 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{3} \,{\left (g^{2} x^{3} e^{12} + 6 \, d g^{2} x^{2} e^{11} + 24 \, d^{2} g^{2} x e^{10} + 3 \, f g x^{2} e^{12} + 24 \, d f g x e^{11} + 3 \, f^{2} x e^{12}\right )} e^{\left (-12\right )} + \frac{2 \,{\left (3 \, d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac{4 \,{\left (d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} + d^{3} f^{2} e^{5} +{\left (d^{4} g^{2} e^{4} + 2 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3*d^3*g^2*e^3 + 4*d^2*f*g*e^4 + d*f^2*e^5)*e^(-6)*ln(abs(x^2*e^2 - d^2)) + 1/
3*(g^2*x^3*e^12 + 6*d*g^2*x^2*e^11 + 24*d^2*g^2*x*e^10 + 3*f*g*x^2*e^12 + 24*d*f
*g*x*e^11 + 3*f^2*x*e^12)*e^(-12) + 2*(3*d^4*g^2*e^4 + 4*d^3*f*g*e^5 + d^2*f^2*e
^6)*e^(-7)*ln(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d) - 4*(d
^5*g^2*e^3 + 2*d^4*f*g*e^4 + d^3*f^2*e^5 + (d^4*g^2*e^4 + 2*d^3*f*g*e^5 + d^2*f^
2*e^6)*x)*e^(-6)/(x^2*e^2 - d^2)