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

Optimal. Leaf size=50 \[ -\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{g x (d g+e f)}{e^2}-\frac{(f+g x)^2}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.0676834, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{(d g+e f)^2 \log (d-e x)}{e^3}-\frac{g x (d g+e f)}{e^2}-\frac{(f+g x)^2}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.032685, size = 43, normalized size = 0.86 \[ -\frac{e g x (2 d g+4 e f+e g x)+2 (d g+e f)^2 \log (d-e x)}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 82, normalized size = 1.6 \[ -{\frac{{g}^{2}{x}^{2}}{2\,e}}-{\frac{{g}^{2}dx}{{e}^{2}}}-2\,{\frac{gfx}{e}}-{\frac{\ln \left ( ex-d \right ){d}^{2}{g}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex-d \right ) dfg}{{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.691315, size = 85, normalized size = 1.7 \[ -\frac{e g^{2} x^{2} + 2 \,{\left (2 \, e f g + d g^{2}\right )} x}{2 \, e^{2}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273369, size = 86, normalized size = 1.72 \[ -\frac{e^{2} g^{2} x^{2} + 2 \,{\left (2 \, e^{2} f g + d e g^{2}\right )} x + 2 \,{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.5712, size = 46, normalized size = 0.92 \[ - \frac{g^{2} x^{2}}{2 e} - \frac{x \left (d g^{2} + 2 e f g\right )}{e^{2}} - \frac{\left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275754, size = 181, normalized size = 3.62 \[ -\frac{1}{2} \,{\left (d^{2} g^{2} e + 2 \, d f g e^{2} + f^{2} e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{2} \,{\left (g^{2} x^{2} e^{3} + 2 \, d g^{2} x e^{2} + 4 \, f g x e^{3}\right )} e^{\left (-4\right )} - \frac{{\left (d^{3} g^{2} + 2 \, d^{2} f g e + d f^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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