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

Optimal. Leaf size=74 \[ \frac{(f+g x) \left (d^2 g+e^2 f x\right )}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}+\frac{(e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0291274, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {723, 208} \[ \frac{(f+g x) \left (d^2 g+e^2 f x\right )}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}+\frac{(e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^2/(d^2 - e^2*x^2)^2,x]

[Out]

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

Rule 723

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\frac{\left (d^2 g+e^2 f x\right ) (f+g x)}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}-\frac{1}{2} \left (-\frac{f^2}{d^2}+\frac{g^2}{e^2}\right ) \int \frac{1}{d^2-e^2 x^2} \, dx\\ &=\frac{\left (d^2 g+e^2 f x\right ) (f+g x)}{2 d^2 e^2 \left (d^2-e^2 x^2\right )}+\frac{(e f-d g) (e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.038201, size = 85, normalized size = 1.15 \[ \frac{-2 d^2 f g-d^2 g^2 x-e^2 f^2 x}{2 d^2 e^2 \left (e^2 x^2-d^2\right )}-\frac{\left (d^2 g^2-e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{2 d^3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.055, size = 180, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}d}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,e{d}^{3}}}-{\frac{{g}^{2}}{4\,{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{2\,d{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{2} \left ( ex-d \right ) }}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}d}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,e{d}^{3}}}-{\frac{{g}^{2}}{4\,{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{2\,d{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{2} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.98369, size = 150, normalized size = 2.03 \begin{align*} -\frac{2 \, d^{2} f g +{\left (e^{2} f^{2} + d^{2} g^{2}\right )} x}{2 \,{\left (d^{2} e^{4} x^{2} - d^{4} e^{2}\right )}} + \frac{{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{3} e^{3}} - \frac{{\left (e^{2} f^{2} - d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{3} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.72916, size = 286, normalized size = 3.86 \begin{align*} -\frac{4 \, d^{3} e f g + 2 \,{\left (d e^{3} f^{2} + d^{3} e g^{2}\right )} x +{\left (d^{2} e^{2} f^{2} - d^{4} g^{2} -{\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x + d\right ) -{\left (d^{2} e^{2} f^{2} - d^{4} g^{2} -{\left (e^{4} f^{2} - d^{2} e^{2} g^{2}\right )} x^{2}\right )} \log \left (e x - d\right )}{4 \,{\left (d^{3} e^{5} x^{2} - d^{5} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.814184, size = 155, normalized size = 2.09 \begin{align*} - \frac{2 d^{2} f g + x \left (d^{2} g^{2} + e^{2} f^{2}\right )}{- 2 d^{4} e^{2} + 2 d^{2} e^{4} x^{2}} + \frac{\left (d g - e f\right ) \left (d g + e f\right ) \log{\left (- \frac{d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} - \frac{\left (d g - e f\right ) \left (d g + e f\right ) \log{\left (\frac{d \left (d g - e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - e^{2} f^{2}\right )} + x \right )}}{4 d^{3} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15209, size = 136, normalized size = 1.84 \begin{align*} \frac{{\left (d^{2} g^{2} - f^{2} e^{2}\right )} e^{\left (-3\right )} \log \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 )}{4 \, d^{2}{\left | d \right |}} - \frac{{\left (d^{2} g^{2} x + 2 \, d^{2} f g + f^{2} x e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x^{2} e^{2} - d^{2}\right )} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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