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

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

[Out]

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

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Rubi [A]  time = 0.0570058, antiderivative size = 50, 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, Rules used = {848, 43} \[ \frac{(d g+e f)^2}{e^3 (d-e x)}+\frac{2 g (d g+e f) \log (d-e x)}{e^3}+\frac{g^2 x}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*x)/e^2 + (e*f + d*g)^2/(e^3*(d - e*x)) + (2*g*(e*f + d*g)*Log[d - e*x])/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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.04354, size = 46, normalized size = 0.92 \[ \frac{\frac{(d g+e f)^2}{d-e x}+2 g (d g+e f) \log (d-e x)+e g^2 x}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 96, normalized size = 1.9 \begin{align*}{\frac{{g}^{2}x}{{e}^{2}}}+2\,{\frac{d\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+2\,{\frac{\ln \left ( ex-d \right ) fg}{{e}^{2}}}-{\frac{{d}^{2}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-2\,{\frac{dfg}{{e}^{2} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{e \left ( ex-d \right ) }} \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)^2,x)

[Out]

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

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Maxima [A]  time = 0.984881, size = 93, normalized size = 1.86 \begin{align*} \frac{g^{2} x}{e^{2}} - \frac{e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}}{e^{4} x - d e^{3}} + \frac{2 \,{\left (e f g + d g^{2}\right )} \log \left (e x - d\right )}{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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70168, size = 184, normalized size = 3.68 \begin{align*} \frac{e^{2} g^{2} x^{2} - d e g^{2} x - e^{2} f^{2} - 2 \, d e f g - d^{2} g^{2} - 2 \,{\left (d e f g + d^{2} g^{2} -{\left (e^{2} f g + d e g^{2}\right )} x\right )} \log \left (e x - d\right )}{e^{4} x - d 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.545168, size = 60, normalized size = 1.2 \begin{align*} - \frac{d^{2} g^{2} + 2 d e f g + e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{g^{2} x}{e^{2}} + \frac{2 g \left (d g + e f\right ) \log{\left (- d + e x \right )}}{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)**2,x)

[Out]

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

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Giac [B]  time = 1.19154, size = 216, normalized size = 4.32 \begin{align*} g^{2} x e^{\left (-2\right )} +{\left (d g^{2} e + f g e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{{\left (d^{2} g^{2} e^{2} + d f g e^{3}\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 )}{{\left | d \right |}} - \frac{{\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3} +{\left (d^{2} g^{2} e^{2} + 2 \, d f g e^{3} + f^{2} e^{4}\right )} x\right )} e^{\left (-4\right )}}{x^{2} e^{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)^2,x, algorithm="giac")

[Out]

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