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

Optimal. Leaf size=109 \[ -\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]

[Out]

-((d*(e*f + 2*d*g)*(3*e*f + 2*d*g)*x)/e^2) - ((e^2*f^2 + 6*d*e*f*g + 4*d^2*g^2)*
x^2)/(2*e) - (g*(2*e*f + 3*d*g)*x^3)/3 - (e*g^2*x^4)/4 - (4*d^2*(e*f + d*g)^2*Lo
g[d - e*x])/e^3

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Rubi [A]  time = 0.242567, antiderivative size = 109, 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 \[ -\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]

Antiderivative was successfully verified.

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

[Out]

-((d*(e*f + 2*d*g)*(3*e*f + 2*d*g)*x)/e^2) - ((e^2*f^2 + 6*d*e*f*g + 4*d^2*g^2)*
x^2)/(2*e) - (g*(2*e*f + 3*d*g)*x^3)/3 - (e*g^2*x^4)/4 - (4*d^2*(e*f + d*g)^2*Lo
g[d - e*x])/e^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 d^{2} \left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - \frac{g x^{3} \left (3 d g + 2 e f\right )}{3} - \frac{\left (4 d^{2} g^{2} + 6 d e f g + e^{2} f^{2}\right ) \int x\, dx}{e} - \frac{\left (2 d g + e f\right ) \left (2 d g + 3 e f\right ) \int d\, dx}{e^{2}} \]

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),x)

[Out]

-4*d**2*(d*g + e*f)**2*log(d - e*x)/e**3 - e*g**2*x**4/4 - g*x**3*(3*d*g + 2*e*f
)/3 - (4*d**2*g**2 + 6*d*e*f*g + e**2*f**2)*Integral(x, x)/e - (2*d*g + e*f)*(2*
d*g + 3*e*f)*Integral(d, x)/e**2

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Mathematica [A]  time = 0.0952753, size = 103, normalized size = 0.94 \[ -\frac{48 d^2 (d g+e f)^2 \log (d-e x)+e x \left (48 d^3 g^2+24 d^2 e g (4 f+g x)+12 d e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )\right )}{12 e^3} \]

Antiderivative was successfully verified.

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

[Out]

-(e*x*(48*d^3*g^2 + 24*d^2*e*g*(4*f + g*x) + 12*d*e^2*(3*f^2 + 3*f*g*x + g^2*x^2
) + e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2)) + 48*d^2*(e*f + d*g)^2*Log[d - e*x])/(1
2*e^3)

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Maple [A]  time = 0.006, size = 145, normalized size = 1.3 \[ -{\frac{e{g}^{2}{x}^{4}}{4}}-{x}^{3}d{g}^{2}-{\frac{2\,e{x}^{3}fg}{3}}-2\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}-3\,{x}^{2}dfg-{\frac{e{x}^{2}{f}^{2}}{2}}-4\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}-8\,{\frac{{d}^{2}fgx}{e}}-3\,d{f}^{2}x-4\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-8\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-4\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.708116, size = 186, normalized size = 1.71 \[ -\frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 4 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{2} f^{2} + 8 \, d^{2} e f g + 4 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac{4 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.281944, size = 188, normalized size = 1.72 \[ -\frac{3 \, e^{4} g^{2} x^{4} + 4 \,{\left (2 \, e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{3} + 6 \,{\left (e^{4} f^{2} + 6 \, d e^{3} f g + 4 \, d^{2} e^{2} g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x + 48 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{12 \, e^{3}} \]

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),x, algorithm="fricas")

[Out]

-1/12*(3*e^4*g^2*x^4 + 4*(2*e^4*f*g + 3*d*e^3*g^2)*x^3 + 6*(e^4*f^2 + 6*d*e^3*f*
g + 4*d^2*e^2*g^2)*x^2 + 12*(3*d*e^3*f^2 + 8*d^2*e^2*f*g + 4*d^3*e*g^2)*x + 48*(
d^2*e^2*f^2 + 2*d^3*e*f*g + d^4*g^2)*log(e*x - d))/e^3

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Sympy [A]  time = 2.2243, size = 116, normalized size = 1.06 \[ - \frac{4 d^{2} \left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - x^{3} \left (d g^{2} + \frac{2 e f g}{3}\right ) - \frac{x^{2} \left (4 d^{2} g^{2} + 6 d e f g + e^{2} f^{2}\right )}{2 e} - \frac{x \left (4 d^{3} g^{2} + 8 d^{2} e f g + 3 d e^{2} f^{2}\right )}{e^{2}} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273402, size = 285, normalized size = 2.61 \[ -2 \,{\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{12} \,{\left (3 \, g^{2} x^{4} e^{9} + 12 \, d g^{2} x^{3} e^{8} + 24 \, d^{2} g^{2} x^{2} e^{7} + 48 \, d^{3} g^{2} x e^{6} + 8 \, f g x^{3} e^{9} + 36 \, d f g x^{2} e^{8} + 96 \, d^{2} f g x e^{7} + 6 \, f^{2} x^{2} e^{9} + 36 \, d f^{2} x e^{8}\right )} e^{\left (-8\right )} - \frac{2 \,{\left (d^{5} g^{2} e^{2} + 2 \, d^{4} f g e^{3} + d^{3} f^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \]

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),x, algorithm="giac")

[Out]

-2*(d^4*g^2*e^3 + 2*d^3*f*g*e^4 + d^2*f^2*e^5)*e^(-6)*ln(abs(x^2*e^2 - d^2)) - 1
/12*(3*g^2*x^4*e^9 + 12*d*g^2*x^3*e^8 + 24*d^2*g^2*x^2*e^7 + 48*d^3*g^2*x*e^6 +
8*f*g*x^3*e^9 + 36*d*f*g*x^2*e^8 + 96*d^2*f*g*x*e^7 + 6*f^2*x^2*e^9 + 36*d*f^2*x
*e^8)*e^(-8) - 2*(d^5*g^2*e^2 + 2*d^4*f*g*e^3 + d^3*f^2*e^4)*e^(-5)*ln(abs(2*x*e
^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d)