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

Optimal. Leaf size=88 \[ \frac{(e f-3 d g) (d g+e f)}{4 d^2 e^3 (d-e x)}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}+\frac{(d g+e f)^2}{4 d e^3 (d-e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0985019, antiderivative size = 88, 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} \[ \frac{(e f-3 d g) (d g+e f)}{4 d^2 e^3 (d-e x)}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}+\frac{(d g+e f)^2}{4 d e^3 (d-e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*f + d*g)^2/(4*d*e^3*(d - e*x)^2) + ((e*f - 3*d*g)*(e*f + d*g))/(4*d^2*e^3*(d - e*x)) + ((e*f - d*g)^2*ArcTa
nh[(e*x)/d])/(4*d^3*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{(d+e x)^2 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^3 (d+e x)} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{2 d e^2 (d-e x)^3}+\frac{(e f-3 d g) (e f+d g)}{4 d^2 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{4 d e^3 (d-e x)^2}+\frac{(e f-3 d g) (e f+d g)}{4 d^2 e^3 (d-e x)}+\frac{(e f-d g)^2 \int \frac{1}{d^2-e^2 x^2} \, dx}{4 d^2 e^2}\\ &=\frac{(e f+d g)^2}{4 d e^3 (d-e x)^2}+\frac{(e f-3 d g) (e f+d g)}{4 d^2 e^3 (d-e x)}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{4 d^3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0810217, size = 90, normalized size = 1.02 \[ \frac{-\frac{2 d (d g+e f) \left (2 d^2 g-d e (2 f+3 g x)+e^2 f x\right )}{(d-e x)^2}+(e f-d g)^2 (-\log (d-e x))+(e f-d g)^2 \log (d+e x)}{8 d^3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.051, size = 218, normalized size = 2.5 \begin{align*}{\frac{3\,{g}^{2}}{4\,{e}^{3} \left ( ex-d \right ) }}+{\frac{fg}{2\,d{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{4\,{d}^{2}e \left ( ex-d \right ) }}+{\frac{{g}^{2}d}{4\,{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{2\,{e}^{2} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{4\,de \left ( ex-d \right ) ^{2}}}-{\frac{\ln \left ( ex-d \right ){g}^{2}}{8\,d{e}^{3}}}+{\frac{\ln \left ( ex-d \right ) fg}{4\,{d}^{2}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{8\,{d}^{3}e}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{8\,d{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) fg}{4\,{d}^{2}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{8\,{d}^{3}e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06183, size = 203, normalized size = 2.31 \begin{align*} \frac{2 \, d e^{2} f^{2} - 2 \, d^{3} g^{2} -{\left (e^{3} f^{2} - 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x}{4 \,{\left (d^{2} e^{5} x^{2} - 2 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{8 \, d^{3} e^{3}} - \frac{{\left (e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{8 \, d^{3} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.80241, size = 547, normalized size = 6.22 \begin{align*} \frac{4 \, d^{2} e^{2} f^{2} - 4 \, d^{4} g^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g - 3 \, d^{3} e g^{2}\right )} x +{\left (d^{2} e^{2} f^{2} - 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x + d\right ) -{\left (d^{2} e^{2} f^{2} - 2 \, d^{3} e f g + d^{4} g^{2} +{\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (d e^{3} f^{2} - 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{8 \,{\left (d^{3} e^{5} x^{2} - 2 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.21287, size = 185, normalized size = 2.1 \begin{align*} \frac{- 2 d^{3} g^{2} + 2 d e^{2} f^{2} + x \left (3 d^{2} e g^{2} + 2 d e^{2} f g - e^{3} f^{2}\right )}{4 d^{4} e^{3} - 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac{\left (d g - e f\right )^{2} \log{\left (- \frac{d \left (d g - e f\right )^{2}}{e \left (d^{2} g^{2} - 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac{\left (d g - e f\right )^{2} \log{\left (\frac{d \left (d g - e f\right )^{2}}{e \left (d^{2} g^{2} - 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15882, size = 266, normalized size = 3.02 \begin{align*} -\frac{{\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} e^{\left (-5\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 )}{8 \, d^{2}{\left | d \right |}} + \frac{{\left (3 \, d^{2} g^{2} x^{3} e^{4} + 4 \, d^{3} g^{2} x^{2} e^{3} - d^{4} g^{2} x e^{2} - 2 \, d^{5} g^{2} e + 2 \, d f g x^{3} e^{5} + 4 \, d^{2} f g x^{2} e^{4} + 2 \, d^{3} f g x e^{3} - f^{2} x^{3} e^{6} + 3 \, d^{2} f^{2} x e^{4} + 2 \, d^{3} f^{2} e^{3}\right )} e^{\left (-4\right )}}{4 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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