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

Optimal. Leaf size=61 \[ -\frac{2 g (d g+e f)}{e^3 (d-e x)}+\frac{(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac{g^2 \log (d-e x)}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.11208, antiderivative size = 61, 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 \[ -\frac{2 g (d g+e f)}{e^3 (d-e x)}+\frac{(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac{g^2 \log (d-e x)}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.7184, size = 51, normalized size = 0.84 \[ - \frac{g^{2} \log{\left (d - e x \right )}}{e^{3}} - \frac{2 g \left (d g + e f\right )}{e^{3} \left (d - e x\right )} + \frac{\left (d g + e f\right )^{2}}{2 e^{3} \left (d - e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0489859, size = 49, normalized size = 0.8 \[ \frac{\frac{(d g+e f) (e (f+4 g x)-3 d g)}{(d-e x)^2}-2 g^2 \log (d-e x)}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 105, normalized size = 1.7 \[ -{\frac{\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+2\,{\frac{{g}^{2}d}{{e}^{3} \left ( ex-d \right ) }}+2\,{\frac{fg}{{e}^{2} \left ( ex-d \right ) }}+{\frac{{d}^{2}{g}^{2}}{2\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fgd}{{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{2\,e \left ( ex-d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.694079, size = 109, normalized size = 1.79 \[ \frac{e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \,{\left (e^{2} f g + d e g^{2}\right )} x}{2 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} - \frac{g^{2} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272137, size = 135, normalized size = 2.21 \[ \frac{e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \,{\left (e^{2} f g + d e g^{2}\right )} x - 2 \,{\left (e^{2} g^{2} x^{2} - 2 \, d e g^{2} x + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.72868, size = 80, normalized size = 1.31 \[ \frac{- 3 d^{2} g^{2} - 2 d e f g + e^{2} f^{2} + x \left (4 d e g^{2} + 4 e^{2} f g\right )}{2 d^{2} e^{3} - 4 d e^{4} x + 2 e^{5} x^{2}} - \frac{g^{2} \log{\left (- d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271113, size = 263, normalized size = 4.31 \[ -\frac{d g^{2} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} - \frac{1}{2} \, g^{2} e^{\left (-3\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{{\left (4 \,{\left (d^{2} g^{2} e^{4} + d f g e^{5}\right )} x^{3} +{\left (5 \, d^{3} g^{2} e^{3} + 6 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} x^{2} - 2 \,{\left (d^{4} g^{2} e^{2} - d^{2} f^{2} e^{4}\right )} x -{\left (3 \, d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} - d^{3} f^{2} e^{5}\right )} e^{\left (-2\right )}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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