3.556 \(\int \frac{(f+g x)^2}{(d+e x)^4 (d^2-e^2 x^2)} \, dx\)

Optimal. Leaf size=139 \[ -\frac{(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac{(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac{(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^5 e^3}-\frac{(e f-d g)^2}{8 d e^3 (d+e x)^4} \]

[Out]

-(e*f - d*g)^2/(8*d*e^3*(d + e*x)^4) - ((e*f - d*g)*(e*f + 3*d*g))/(12*d^2*e^3*(d + e*x)^3) - (e*f + d*g)^2/(1
6*d^3*e^3*(d + e*x)^2) - (e*f + d*g)^2/(16*d^4*e^3*(d + e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(16*d^5*e^3)

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Rubi [A]  time = 0.133073, antiderivative size = 139, 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} \[ -\frac{(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac{(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac{(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}+\frac{(d g+e f)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^5 e^3}-\frac{(e f-d g)^2}{8 d e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f - d*g)^2/(8*d*e^3*(d + e*x)^4) - ((e*f - d*g)*(e*f + 3*d*g))/(12*d^2*e^3*(d + e*x)^3) - (e*f + d*g)^2/(1
6*d^3*e^3*(d + e*x)^2) - (e*f + d*g)^2/(16*d^4*e^3*(d + e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(16*d^5*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)^4 \left (d^2-e^2 x^2\right )} \, dx &=\int \frac{(f+g x)^2}{(d-e x) (d+e x)^5} \, dx\\ &=\int \left (\frac{(-e f+d g)^2}{2 d e^2 (d+e x)^5}+\frac{(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^4}+\frac{(e f+d g)^2}{8 d^3 e^2 (d+e x)^3}+\frac{(e f+d g)^2}{16 d^4 e^2 (d+e x)^2}+\frac{(e f+d g)^2}{16 d^4 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac{(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac{(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac{(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac{(e f+d g)^2 \int \frac{1}{d^2-e^2 x^2} \, dx}{16 d^4 e^2}\\ &=-\frac{(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac{(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac{(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac{(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac{(e f+d g)^2 \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^5 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0940853, size = 142, normalized size = 1.02 \[ -\frac{\frac{8 d^3 \left (-3 d^2 g^2+2 d e f g+e^2 f^2\right )}{(d+e x)^3}+\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d^2 (d g+e f)^2}{(d+e x)^2}+\frac{6 d (d g+e f)^2}{d+e x}+3 (d g+e f)^2 \log (d-e x)-3 (d g+e f)^2 \log (d+e x)}{96 d^5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (8*d^3*(e^2*f^2 + 2*d*e*f*g - 3*d^2*g^2))/(d + e*x)^3 + (6*d^2*(e*f + d
*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])/(96*d^5*e^3)

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Maple [B]  time = 0.056, size = 312, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{32\,{d}^{3}{e}^{3}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{d}^{4}{e}^{2}}}-{\frac{\ln \left ( ex-d \right ){f}^{2}}{32\,{d}^{5}e}}+{\frac{{g}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{fg}{6\,d{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{12\,{d}^{2}e \left ( ex+d \right ) ^{3}}}-{\frac{{g}^{2}d}{8\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{8\,de \left ( ex+d \right ) ^{4}}}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{32\,{d}^{3}{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{d}^{4}{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){f}^{2}}{32\,{d}^{5}e}}-{\frac{{g}^{2}}{16\,{d}^{2}{e}^{3} \left ( ex+d \right ) }}-{\frac{fg}{8\,{d}^{3}{e}^{2} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{16\,{d}^{4}e \left ( ex+d \right ) }}-{\frac{{g}^{2}}{16\,d{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{fg}{8\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{16\,{d}^{3}e \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)^4/(-e^2*x^2+d^2),x)

[Out]

-1/32/e^3/d^3*ln(e*x-d)*g^2-1/16/e^2/d^4*ln(e*x-d)*f*g-1/32/e/d^5*ln(e*x-d)*f^2+1/4/e^3/(e*x+d)^3*g^2-1/6/d/e^
2/(e*x+d)^3*f*g-1/12/d^2/e/(e*x+d)^3*f^2-1/8*d/e^3/(e*x+d)^4*g^2+1/4/e^2/(e*x+d)^4*f*g-1/8/d/e/(e*x+d)^4*f^2+1
/32/e^3/d^3*ln(e*x+d)*g^2+1/16/e^2/d^4*ln(e*x+d)*f*g+1/32/e/d^5*ln(e*x+d)*f^2-1/16/d^2/e^3/(e*x+d)*g^2-1/8/d^3
/e^2/(e*x+d)*f*g-1/16/d^4/e/(e*x+d)*f^2-1/16/d/e^3/(e*x+d)^2*g^2-1/8/d^2/e^2/(e*x+d)^2*f*g-1/16/d^3/e/(e*x+d)^
2*f^2

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Maxima [A]  time = 1.04339, size = 319, normalized size = 2.29 \begin{align*} -\frac{16 \, d^{3} e f^{2} + 8 \, d^{4} f g + 3 \,{\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{3} + 12 \,{\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x^{2} +{\left (19 \, d^{2} e^{2} f^{2} + 38 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} x}{48 \,{\left (d^{4} e^{6} x^{4} + 4 \, d^{5} e^{5} x^{3} + 6 \, d^{6} e^{4} x^{2} + 4 \, d^{7} e^{3} x + d^{8} e^{2}\right )}} + \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{5} e^{3}} - \frac{{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{5} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="maxima")

[Out]

-1/48*(16*d^3*e*f^2 + 8*d^4*f*g + 3*(e^4*f^2 + 2*d*e^3*f*g + d^2*e^2*g^2)*x^3 + 12*(d*e^3*f^2 + 2*d^2*e^2*f*g
+ d^3*e*g^2)*x^2 + (19*d^2*e^2*f^2 + 38*d^3*e*f*g + 3*d^4*g^2)*x)/(d^4*e^6*x^4 + 4*d^5*e^5*x^3 + 6*d^6*e^4*x^2
+ 4*d^7*e^3*x + d^8*e^2) + 1/32*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x + d)/(d^5*e^3) - 1/32*(e^2*f^2 + 2*d*
e*f*g + d^2*g^2)*log(e*x - d)/(d^5*e^3)

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Fricas [B]  time = 1.75234, size = 1031, normalized size = 7.42 \begin{align*} -\frac{32 \, d^{4} e^{2} f^{2} + 16 \, d^{5} e f g + 6 \,{\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 24 \,{\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x - 3 \,{\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} +{\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \,{\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \,{\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \,{\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} +{\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \,{\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \,{\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \,{\left (d^{5} e^{7} x^{4} + 4 \, d^{6} e^{6} x^{3} + 6 \, d^{7} e^{5} x^{2} + 4 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="fricas")

[Out]

-1/96*(32*d^4*e^2*f^2 + 16*d^5*e*f*g + 6*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3*e^3*g^2)*x^3 + 24*(d^2*e^4*f^2 + 2*d
^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 2*(19*d^3*e^3*f^2 + 38*d^4*e^2*f*g + 3*d^5*e*g^2)*x - 3*(d^4*e^2*f^2 + 2*d^5*e
*f*g + d^6*g^2 + (e^6*f^2 + 2*d*e^5*f*g + d^2*e^4*g^2)*x^4 + 4*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3*e^3*g^2)*x^3 +
6*(d^2*e^4*f^2 + 2*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 4*(d^3*e^3*f^2 + 2*d^4*e^2*f*g + d^5*e*g^2)*x)*log(e*x +
d) + 3*(d^4*e^2*f^2 + 2*d^5*e*f*g + d^6*g^2 + (e^6*f^2 + 2*d*e^5*f*g + d^2*e^4*g^2)*x^4 + 4*(d*e^5*f^2 + 2*d^2
*e^4*f*g + d^3*e^3*g^2)*x^3 + 6*(d^2*e^4*f^2 + 2*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 4*(d^3*e^3*f^2 + 2*d^4*e^2*f
*g + d^5*e*g^2)*x)*log(e*x - d))/(d^5*e^7*x^4 + 4*d^6*e^6*x^3 + 6*d^7*e^5*x^2 + 4*d^8*e^4*x + d^9*e^3)

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Sympy [B]  time = 1.69364, size = 282, normalized size = 2.03 \begin{align*} - \frac{8 d^{4} f g + 16 d^{3} e f^{2} + x^{3} \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x^{2} \left (12 d^{3} e g^{2} + 24 d^{2} e^{2} f g + 12 d e^{3} f^{2}\right ) + x \left (3 d^{4} g^{2} + 38 d^{3} e f g + 19 d^{2} e^{2} f^{2}\right )}{48 d^{8} e^{2} + 192 d^{7} e^{3} x + 288 d^{6} e^{4} x^{2} + 192 d^{5} e^{5} x^{3} + 48 d^{4} e^{6} x^{4}} - \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 )}}{32 d^{5} 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 )}}{32 d^{5} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2/(e*x+d)**4/(-e**2*x**2+d**2),x)

[Out]

-(8*d**4*f*g + 16*d**3*e*f**2 + x**3*(3*d**2*e**2*g**2 + 6*d*e**3*f*g + 3*e**4*f**2) + x**2*(12*d**3*e*g**2 +
24*d**2*e**2*f*g + 12*d*e**3*f**2) + x*(3*d**4*g**2 + 38*d**3*e*f*g + 19*d**2*e**2*f**2))/(48*d**8*e**2 + 192*
d**7*e**3*x + 288*d**6*e**4*x**2 + 192*d**5*e**5*x**3 + 48*d**4*e**6*x**4) - (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)/(32*d**5*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)/(32*d**5*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)^4/(-e^2*x^2+d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError