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

Optimal. Leaf size=113 \[ -\frac{(3 d g+e f) (e f-d g)}{8 d^2 e^3 (d+e x)^2}-\frac{(d g+e f)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}-\frac{(e f-d g)^2}{6 d e^3 (d+e x)^3} \]

[Out]

-(e*f - d*g)^2/(6*d*e^3*(d + e*x)^3) - ((e*f - d*g)*(e*f + 3*d*g))/(8*d^2*e^3*(d + e*x)^2) - (e*f + d*g)^2/(8*
d^3*e^3*(d + e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(8*d^4*e^3)

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Rubi [A]  time = 0.115919, antiderivative size = 113, 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} \[ -\frac{(3 d g+e f) (e f-d g)}{8 d^2 e^3 (d+e x)^2}-\frac{(d g+e f)^2}{8 d^3 e^3 (d+e x)}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}-\frac{(e f-d g)^2}{6 d e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f - d*g)^2/(6*d*e^3*(d + e*x)^3) - ((e*f - d*g)*(e*f + 3*d*g))/(8*d^2*e^3*(d + e*x)^2) - (e*f + d*g)^2/(8*
d^3*e^3*(d + e*x)) + ((e*f + d*g)^2*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)^3 \left (d^2-e^2 x^2\right )} \, dx &=\int \frac{(f+g x)^2}{(d-e x) (d+e x)^4} \, dx\\ &=\int \left (\frac{(-e f+d g)^2}{2 d e^2 (d+e x)^4}+\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^3}+\frac{(e f+d g)^2}{8 d^3 e^2 (d+e x)^2}+\frac{(e f+d g)^2}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{(e f-d g)^2}{6 d e^3 (d+e x)^3}-\frac{(e f-d g) (e f+3 d g)}{8 d^2 e^3 (d+e x)^2}-\frac{(e f+d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(e f+d g)^2 \int \frac{1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=-\frac{(e f-d g)^2}{6 d e^3 (d+e x)^3}-\frac{(e f-d g) (e f+3 d g)}{8 d^2 e^3 (d+e x)^2}-\frac{(e f+d g)^2}{8 d^3 e^3 (d+e x)}+\frac{(e f+d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{8 d^4 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0583331, size = 122, normalized size = 1.08 \[ \frac{\frac{6 d^2 \left (3 d^2 g^2-2 d e f g-e^2 f^2\right )}{(d+e x)^2}-\frac{8 d^3 (e f-d g)^2}{(d+e x)^3}-\frac{6 d (d g+e f)^2}{d+e x}-3 (d g+e f)^2 \log (d-e x)+3 (d g+e f)^2 \log (d+e x)}{48 d^4 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*d^3*(e*f - d*g)^2)/(d + e*x)^3 + (6*d^2*(-(e^2*f^2) - 2*d*e*f*g + 3*d^2*g^2))/(d + e*x)^2 - (6*d*(e*f + d
*g)^2)/(d + e*x) - 3*(e*f + d*g)^2*Log[d - e*x] + 3*(e*f + d*g)^2*Log[d + e*x])/(48*d^4*e^3)

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Maple [B]  time = 0.054, size = 259, normalized size = 2.3 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{16\,{d}^{2}{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{8\,{d}^{3}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{16\,{d}^{4}e}}+{\frac{3\,{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\,{d}^{2}e \left ( ex+d \right ) ^{2}}}-{\frac{{g}^{2}d}{6\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{6\,de \left ( ex+d \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{16\,{d}^{2}{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{8\,{d}^{3}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{16\,{d}^{4}e}}-{\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\,{d}^{3}e \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)^3/(-e^2*x^2+d^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-1/16/e/d^4*ln(e*x-d)*f^2+3/8/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^2-1/6*d/e^3/(e*x+d)^3*g^2+1/3/e^2/(e*x+d)^3*f*g-1/6/d/e/(e*x+d)^3*f^2+1/1
6/e^3/d^2*ln(e*x+d)*g^2+1/8/e^2/d^3*ln(e*x+d)*f*g+1/16/e/d^4*ln(e*x+d)*f^2-1/8/d/e^3/(e*x+d)*g^2-1/4/d^2/e^2/(
e*x+d)*f*g-1/8/d^3/e/(e*x+d)*f^2

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Maxima [A]  time = 0.992291, size = 278, normalized size = 2.46 \begin{align*} -\frac{10 \, d^{2} e^{2} f^{2} + 4 \, d^{3} e f g - 2 \, d^{4} g^{2} + 3 \,{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (3 \, d e^{3} f^{2} + 6 \, d^{2} e^{2} f g - d^{3} e g^{2}\right )} x}{24 \,{\left (d^{3} e^{6} x^{3} + 3 \, d^{4} e^{5} x^{2} + 3 \, d^{5} e^{4} x + d^{6} 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 )}{16 \, d^{4} e^{3}} - \frac{{\left (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)^3/(-e^2*x^2+d^2),x, algorithm="maxima")

[Out]

-1/24*(10*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*(3*d*e^3*f^2
+ 6*d^2*e^2*f*g - d^3*e*g^2)*x)/(d^3*e^6*x^3 + 3*d^4*e^5*x^2 + 3*d^5*e^4*x + d^6*e^3) + 1/16*(e^2*f^2 + 2*d*e
*f*g + d^2*g^2)*log(e*x + d)/(d^4*e^3) - 1/16*(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 = 1.80371, size = 807, normalized size = 7.14 \begin{align*} -\frac{20 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g - 4 \, d^{5} g^{2} + 6 \,{\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 6 \,{\left (3 \, d^{2} e^{3} f^{2} + 6 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x - 3 \,{\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2} +{\left (e^{5} f^{2} + 2 \, d e^{4} f g + d^{2} e^{3} g^{2}\right )} x^{3} + 3 \,{\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (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 ) + 3 \,{\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2} +{\left (e^{5} f^{2} + 2 \, d e^{4} f g + d^{2} e^{3} g^{2}\right )} x^{3} + 3 \,{\left (d e^{4} f^{2} + 2 \, d^{2} e^{3} f g + d^{3} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (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 )}{48 \,{\left (d^{4} e^{6} x^{3} + 3 \, d^{5} e^{5} x^{2} + 3 \, 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)^3/(-e^2*x^2+d^2),x, algorithm="fricas")

[Out]

-1/48*(20*d^3*e^2*f^2 + 8*d^4*e*f*g - 4*d^5*g^2 + 6*(d*e^4*f^2 + 2*d^2*e^3*f*g + d^3*e^2*g^2)*x^2 + 6*(3*d^2*e
^3*f^2 + 6*d^3*e^2*f*g - d^4*e*g^2)*x - 3*(d^3*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2 + (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 + (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 + 3*d^5*e^5*x^2 + 3*d^6*e^4*x + d^7*e^3)

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Sympy [B]  time = 1.40659, size = 248, normalized size = 2.19 \begin{align*} - \frac{- 2 d^{4} g^{2} + 4 d^{3} e f g + 10 d^{2} e^{2} f^{2} + x^{2} \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} + 18 d^{2} e^{2} f g + 9 d e^{3} f^{2}\right )}{24 d^{6} e^{3} + 72 d^{5} e^{4} x + 72 d^{4} e^{5} x^{2} + 24 d^{3} e^{6} x^{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 )}}{16 d^{4} 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 )}}{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)**3/(-e**2*x**2+d**2),x)

[Out]

-(-2*d**4*g**2 + 4*d**3*e*f*g + 10*d**2*e**2*f**2 + x**2*(3*d**2*e**2*g**2 + 6*d*e**3*f*g + 3*e**4*f**2) + x*(
-3*d**3*e*g**2 + 18*d**2*e**2*f*g + 9*d*e**3*f**2))/(24*d**6*e**3 + 72*d**5*e**4*x + 72*d**4*e**5*x**2 + 24*d*
*3*e**6*x**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)/(16*d**4*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)/(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)^3/(-e^2*x^2+d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError