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

Optimal. Leaf size=121 \[ -\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}-\frac{(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}+\frac{(d g+e f)^2}{8 d^3 e^3 (d-e x)}+\frac{(3 e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3} \]

[Out]

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

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Rubi [A]  time = 0.135801, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {848, 88, 208} \[ -\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}-\frac{(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}+\frac{(d g+e f)^2}{8 d^3 e^3 (d-e x)}+\frac{(3 e f-d g) (d g+e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -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}{(d+e x) \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)^3} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{8 d^3 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)^3}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^2}+\frac{(3 e f-d g) (e f+d g)}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{8 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}+\frac{((3 e f-d g) (e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=\frac{(e f+d g)^2}{8 d^3 e^3 (d-e x)}-\frac{(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac{e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}+\frac{(3 e f-d g) (e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}\\ \end{align*}

Mathematica [A]  time = 0.104534, size = 139, normalized size = 1.15 \[ \frac{\frac{4 d \left (d^2 g^2-e^2 f^2\right )}{d+e x}+\left (d^2 g^2-2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2+2 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac{2 d^2 (e f-d g)^2}{(d+e x)^2}+\frac{2 d (d g+e f)^2}{d-e x}}{16 d^4 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*d*(e*f + d*g)^2)/(d - e*x) - (2*d^2*(e*f - d*g)^2)/(d + e*x)^2 + (4*d*(-(e^2*f^2) + d^2*g^2))/(d + e*x) +
(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + (3*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(16*d^4*e^3)

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Maple [B]  time = 0.055, size = 253, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( ex-d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}-{\frac{\ln \left ( ex-d \right ) fg}{8\,{e}^{2}{d}^{3}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{{g}^{2}}{8\,d{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{4\,{d}^{2}{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{3} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{4\,d{e}^{3} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{4\,e{d}^{3} \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{16\,{e}^{3}{d}^{2}}}+{\frac{\ln \left ( ex+d \right ) fg}{8\,{e}^{2}{d}^{3}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{16\,e{d}^{4}}}-{\frac{{g}^{2}}{8\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{fg}{4\,d{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{8\,e{d}^{2} \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.982601, size = 286, normalized size = 2.36 \begin{align*} \frac{2 \, d^{2} e^{2} f^{2} - 4 \, d^{3} e f g - 2 \, d^{4} g^{2} -{\left (3 \, e^{4} f^{2} + 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} -{\left (3 \, d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \,{\left (d^{3} e^{6} x^{3} + d^{4} e^{5} x^{2} - d^{5} e^{4} x - d^{6} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*d^2*e^2*f^2 - 4*d^3*e*f*g - 2*d^4*g^2 - (3*e^4*f^2 + 2*d*e^3*f*g - d^2*e^2*g^2)*x^2 - (3*d*e^3*f^2 + 2*
d^2*e^2*f*g + 3*d^3*e*g^2)*x)/(d^3*e^6*x^3 + d^4*e^5*x^2 - d^5*e^4*x - d^6*e^3) + 1/16*(3*e^2*f^2 + 2*d*e*f*g
- d^2*g^2)*log(e*x + d)/(d^4*e^3) - 1/16*(3*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(e*x - d)/(d^4*e^3)

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Fricas [B]  time = 1.80002, size = 810, normalized size = 6.69 \begin{align*} \frac{4 \, d^{3} e^{2} f^{2} - 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \,{\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x -{\left (3 \, d^{3} e^{2} f^{2} + 2 \, d^{4} e f g - d^{5} g^{2} -{\left (3 \, e^{5} f^{2} + 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} +{\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (3 \, d^{3} e^{2} f^{2} + 2 \, d^{4} e f g - d^{5} g^{2} -{\left (3 \, e^{5} f^{2} + 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} -{\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} +{\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{16 \,{\left (d^{4} e^{6} x^{3} + d^{5} e^{5} x^{2} - d^{6} e^{4} x - d^{7} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*d^3*e^2*f^2 - 8*d^4*e*f*g - 4*d^5*g^2 - 2*(3*d*e^4*f^2 + 2*d^2*e^3*f*g - d^3*e^2*g^2)*x^2 - 2*(3*d^2*e
^3*f^2 + 2*d^3*e^2*f*g + 3*d^4*e*g^2)*x - (3*d^3*e^2*f^2 + 2*d^4*e*f*g - d^5*g^2 - (3*e^5*f^2 + 2*d*e^4*f*g -
d^2*e^3*g^2)*x^3 - (3*d*e^4*f^2 + 2*d^2*e^3*f*g - d^3*e^2*g^2)*x^2 + (3*d^2*e^3*f^2 + 2*d^3*e^2*f*g - d^4*e*g^
2)*x)*log(e*x + d) + (3*d^3*e^2*f^2 + 2*d^4*e*f*g - d^5*g^2 - (3*e^5*f^2 + 2*d*e^4*f*g - d^2*e^3*g^2)*x^3 - (3
*d*e^4*f^2 + 2*d^2*e^3*f*g - d^3*e^2*g^2)*x^2 + (3*d^2*e^3*f^2 + 2*d^3*e^2*f*g - d^4*e*g^2)*x)*log(e*x - d))/(
d^4*e^6*x^3 + d^5*e^5*x^2 - d^6*e^4*x - d^7*e^3)

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Sympy [B]  time = 1.50707, size = 279, normalized size = 2.31 \begin{align*} \frac{- 2 d^{4} g^{2} - 4 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x^{2} \left (d^{2} e^{2} g^{2} - 2 d e^{3} f g - 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} - 2 d^{2} e^{2} f g - 3 d e^{3} f^{2}\right )}{- 8 d^{6} e^{3} - 8 d^{5} e^{4} x + 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac{\left (d g - 3 e f\right ) \left (d g + e f\right ) \log{\left (- \frac{d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac{\left (d g - 3 e f\right ) \left (d g + e f\right ) \log{\left (\frac{d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*d**4*g**2 - 4*d**3*e*f*g + 2*d**2*e**2*f**2 + x**2*(d**2*e**2*g**2 - 2*d*e**3*f*g - 3*e**4*f**2) + x*(-3*d
**3*e*g**2 - 2*d**2*e**2*f*g - 3*d*e**3*f**2))/(-8*d**6*e**3 - 8*d**5*e**4*x + 8*d**4*e**5*x**2 + 8*d**3*e**6*
x**3) + (d*g - 3*e*f)*(d*g + e*f)*log(-d*(d*g - 3*e*f)*(d*g + e*f)/(e*(d**2*g**2 - 2*d*e*f*g - 3*e**2*f**2)) +
x)/(16*d**4*e**3) - (d*g - 3*e*f)*(d*g + e*f)*log(d*(d*g - 3*e*f)*(d*g + e*f)/(e*(d**2*g**2 - 2*d*e*f*g - 3*e
**2*f**2)) + x)/(16*d**4*e**3)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError