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

Optimal. Leaf size=86 \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.0855682, antiderivative size = 86, 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, 88} \[ \frac{(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac{(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac{(e f-d g)^2}{2 d e^3 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )} \, dx &=\int \frac{(f+g x)^2}{(d-e x) (d+e x)^2} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{4 d^2 e^2 (d-e x)}+\frac{(-e f+d g)^2}{2 d e^2 (d+e x)^2}+\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)}\right ) \, dx\\ &=-\frac{(e f-d g)^2}{2 d e^3 (d+e x)}-\frac{(e f+d g)^2 \log (d-e x)}{4 d^2 e^3}+\frac{(e f-d g) (e f+3 d g) \log (d+e x)}{4 d^2 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0486066, size = 82, normalized size = 0.95 \[ \frac{(e f-d g) ((d+e x) (3 d g+e f) \log (d+e x)+2 d (d g-e f))-(d+e x) (d g+e f)^2 \log (d-e x)}{4 d^2 e^3 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 149, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{4\,{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{2\,d{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{3\,\ln \left ( ex+d \right ){g}^{2}}{4\,{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{2\,d{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{4\,{d}^{2}e}}-{\frac{{g}^{2}d}{2\,{e}^{3} \left ( ex+d \right ) }}+{\frac{fg}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{2\,de \left ( ex+d \right ) }} \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),x)

[Out]

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

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Maxima [A]  time = 0.985877, size = 153, normalized size = 1.78 \begin{align*} -\frac{e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}}{2 \,{\left (d e^{4} x + d^{2} e^{3}\right )}} + \frac{{\left (e^{2} f^{2} + 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} 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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.83723, size = 343, normalized size = 3.99 \begin{align*} -\frac{2 \, d e^{2} f^{2} - 4 \, d^{2} e f g + 2 \, d^{3} g^{2} -{\left (d e^{2} f^{2} + 2 \, d^{2} e f g - 3 \, d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) +{\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2} +{\left (e^{3} f^{2} + 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \,{\left (d^{2} e^{4} x + d^{3} 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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.14778, size = 182, normalized size = 2.12 \begin{align*} - \frac{d^{2} g^{2} - 2 d e f g + e^{2} f^{2}}{2 d^{2} e^{3} + 2 d e^{4} x} - \frac{\left (d g - e f\right ) \left (3 d g + e f\right ) \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g - e f\right ) \left (3 d g + e f\right )}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} - \frac{\left (d g + e f\right )^{2} \log{\left (x + \frac{- 2 d^{3} g^{2} + d \left (d g + e f\right )^{2}}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} 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),x)

[Out]

-(d**2*g**2 - 2*d*e*f*g + e**2*f**2)/(2*d**2*e**3 + 2*d*e**4*x) - (d*g - e*f)*(3*d*g + e*f)*log(x + (-2*d**3*g
**2 + d*(d*g - e*f)*(3*d*g + e*f))/(d**2*e*g**2 - 2*d*e**2*f*g - e**3*f**2))/(4*d**2*e**3) - (d*g + e*f)**2*lo
g(x + (-2*d**3*g**2 + d*(d*g + e*f)**2)/(d**2*e*g**2 - 2*d*e**2*f*g - e**3*f**2))/(4*d**2*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),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError