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

Optimal. Leaf size=210 \[ -\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}+\frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3} \]

[Out]

(e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) - (e^2*f^2 - d^2*g^2)/(16*d^3*e^
3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(d + e*x)^
2) - ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3*(d + e*x)) + ((e*f + d*g)*(3*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^7
*e^3)

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Rubi [A]  time = 0.241934, antiderivative size = 210, 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} \[ -\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}+\frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) - (e^2*f^2 - d^2*g^2)/(16*d^3*e^
3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(d + e*x)^
2) - ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3*(d + e*x)) + ((e*f + d*g)*(3*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^7
*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 )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)^6} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{64 d^6 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)^6}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^5}+\frac{(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac{f (e f+d g)}{8 d^5 e (d+e x)^3}+\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^2 (d+e x)^2}+\frac{(e f+d g) (3 e f+d g)}{32 d^6 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac{f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac{((e f+d g) (3 e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{32 d^6 e^2}\\ &=\frac{(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac{f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac{(e f+d g) (3 e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3}\\ \end{align*}

Mathematica [A]  time = 0.18319, size = 229, normalized size = 1.09 \[ \frac{\frac{60 d^4 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^4}+\frac{20 d^3 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}-\frac{15 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d+e x}-15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d-e x)+15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac{48 d^5 (e f-d g)^2}{(d+e x)^5}-\frac{60 d^2 e f (d g+e f)}{(d+e x)^2}+\frac{15 d (d g+e f)^2}{d-e x}}{960 d^7 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((15*d*(e*f + d*g)^2)/(d - e*x) - (48*d^5*(e*f - d*g)^2)/(d + e*x)^5 + (60*d^4*(-(e^2*f^2) + d^2*g^2))/(d + e*
x)^4 + (20*d^3*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^3 - (60*d^2*e*f*(e*f + d*g))/(d + e*x)^2 - (15*d*
(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d + e*x) - 15*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d - e*x] + 15*(3*e^2*f
^2 + 4*d*e*f*g + d^2*g^2)*Log[d + e*x])/(960*d^7*e^3)

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Maple [B]  time = 0.062, size = 394, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{6}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{32\,{e}^{2}{d}^{5} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{64\,e{d}^{6} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{6}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex+d \right ) }}-{\frac{3\,fg}{32\,{e}^{2}{d}^{5} \left ( ex+d \right ) }}-{\frac{5\,{f}^{2}}{64\,e{d}^{6} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}d \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{g}^{2}}{48\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{fg}{24\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{16\,e{d}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{g}^{2}}{20\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{fg}{10\,d{e}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{{f}^{2}}{20\,e{d}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{16\,e{d}^{5} \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)^2,x)

[Out]

-1/64/e^3/d^5*ln(e*x-d)*g^2-1/16/e^2/d^6*ln(e*x-d)*f*g-3/64/e/d^7*ln(e*x-d)*f^2-1/64/e^3/d^4/(e*x-d)*g^2-1/32/
e^2/d^5/(e*x-d)*f*g-1/64/e/d^6/(e*x-d)*f^2+1/64/e^3/d^5*ln(e*x+d)*g^2+1/16/e^2/d^6*ln(e*x+d)*f*g+3/64/e/d^7*ln
(e*x+d)*f^2-1/64/e^3/d^4/(e*x+d)*g^2-3/32/e^2/d^5/(e*x+d)*f*g-5/64/e/d^6/(e*x+d)*f^2+1/16/e^3/d/(e*x+d)^4*g^2-
1/16/e/d^3/(e*x+d)^4*f^2+1/48/e^3/d^2/(e*x+d)^3*g^2-1/24/e^2/d^3/(e*x+d)^3*f*g-1/16/e/d^4/(e*x+d)^3*f^2-1/20/e
^3/(e*x+d)^5*g^2+1/10/d/e^2/(e*x+d)^5*f*g-1/20/d^2/e/(e*x+d)^5*f^2-1/16/e^2*f/d^4/(e*x+d)^2*g-1/16/e*f^2/d^5/(
e*x+d)^2

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Maxima [A]  time = 1.08751, size = 462, normalized size = 2.2 \begin{align*} \frac{144 \, d^{5} e^{2} f^{2} + 32 \, d^{6} e f g - 16 \, d^{7} g^{2} - 15 \,{\left (3 \, e^{7} f^{2} + 4 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 60 \,{\left (3 \, d e^{6} f^{2} + 4 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 80 \,{\left (3 \, d^{2} e^{5} f^{2} + 4 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 20 \,{\left (3 \, d^{3} e^{4} f^{2} + 4 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} +{\left (141 \, d^{4} e^{3} f^{2} + 188 \, d^{5} e^{2} f g - 49 \, d^{6} e g^{2}\right )} x}{480 \,{\left (d^{6} e^{9} x^{6} + 4 \, d^{7} e^{8} x^{5} + 5 \, d^{8} e^{7} x^{4} - 5 \, d^{10} e^{5} x^{2} - 4 \, d^{11} e^{4} x - d^{12} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{7} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{7} 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)^2,x, algorithm="maxima")

[Out]

1/480*(144*d^5*e^2*f^2 + 32*d^6*e*f*g - 16*d^7*g^2 - 15*(3*e^7*f^2 + 4*d*e^6*f*g + d^2*e^5*g^2)*x^5 - 60*(3*d*
e^6*f^2 + 4*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 80*(3*d^2*e^5*f^2 + 4*d^3*e^4*f*g + d^4*e^3*g^2)*x^3 - 20*(3*d^3*
e^4*f^2 + 4*d^4*e^3*f*g + d^5*e^2*g^2)*x^2 + (141*d^4*e^3*f^2 + 188*d^5*e^2*f*g - 49*d^6*e*g^2)*x)/(d^6*e^9*x^
6 + 4*d^7*e^8*x^5 + 5*d^8*e^7*x^4 - 5*d^10*e^5*x^2 - 4*d^11*e^4*x - d^12*e^3) + 1/64*(3*e^2*f^2 + 4*d*e*f*g +
d^2*g^2)*log(e*x + d)/(d^7*e^3) - 1/64*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^7*e^3)

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Fricas [B]  time = 2.13187, size = 1413, normalized size = 6.73 \begin{align*} \frac{288 \, d^{6} e^{2} f^{2} + 64 \, d^{7} e f g - 32 \, d^{8} g^{2} - 30 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 120 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} - 160 \,{\left (3 \, d^{3} e^{5} f^{2} + 4 \, d^{4} e^{4} f g + d^{5} e^{3} g^{2}\right )} x^{3} - 40 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (141 \, d^{5} e^{3} f^{2} + 188 \, d^{6} e^{2} f g - 49 \, d^{7} e g^{2}\right )} x - 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{960 \,{\left (d^{7} e^{9} x^{6} + 4 \, d^{8} e^{8} x^{5} + 5 \, d^{9} e^{7} x^{4} - 5 \, d^{11} e^{5} x^{2} - 4 \, d^{12} e^{4} x - d^{13} 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)^2,x, algorithm="fricas")

[Out]

1/960*(288*d^6*e^2*f^2 + 64*d^7*e*f*g - 32*d^8*g^2 - 30*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 120*
(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 - 160*(3*d^3*e^5*f^2 + 4*d^4*e^4*f*g + d^5*e^3*g^2)*x^3 - 40
*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e^2*g^2)*x^2 + 2*(141*d^5*e^3*f^2 + 188*d^6*e^2*f*g - 49*d^7*e*g^2)*x -
15*(3*d^6*e^2*f^2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g + d^2*e^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d
^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 5*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5
*e^3*f*g + d^6*e^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x + d) + 15*(3*d^6*e^2*f^
2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g + d^2*e^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3
*e^5*g^2)*x^5 - 5*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e
^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x - d))/(d^7*e^9*x^6 + 4*d^8*e^8*x^5 + 5*
d^9*e^7*x^4 - 5*d^11*e^5*x^2 - 4*d^12*e^4*x - d^13*e^3)

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Sympy [B]  time = 2.58207, size = 420, normalized size = 2. \begin{align*} - \frac{16 d^{7} g^{2} - 32 d^{6} e f g - 144 d^{5} e^{2} f^{2} + x^{5} \left (15 d^{2} e^{5} g^{2} + 60 d e^{6} f g + 45 e^{7} f^{2}\right ) + x^{4} \left (60 d^{3} e^{4} g^{2} + 240 d^{2} e^{5} f g + 180 d e^{6} f^{2}\right ) + x^{3} \left (80 d^{4} e^{3} g^{2} + 320 d^{3} e^{4} f g + 240 d^{2} e^{5} f^{2}\right ) + x^{2} \left (20 d^{5} e^{2} g^{2} + 80 d^{4} e^{3} f g + 60 d^{3} e^{4} f^{2}\right ) + x \left (49 d^{6} e g^{2} - 188 d^{5} e^{2} f g - 141 d^{4} e^{3} f^{2}\right )}{- 480 d^{12} e^{3} - 1920 d^{11} e^{4} x - 2400 d^{10} e^{5} x^{2} + 2400 d^{8} e^{7} x^{4} + 1920 d^{7} e^{8} x^{5} + 480 d^{6} e^{9} x^{6}} - \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} 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)**2,x)

[Out]

-(16*d**7*g**2 - 32*d**6*e*f*g - 144*d**5*e**2*f**2 + x**5*(15*d**2*e**5*g**2 + 60*d*e**6*f*g + 45*e**7*f**2)
+ x**4*(60*d**3*e**4*g**2 + 240*d**2*e**5*f*g + 180*d*e**6*f**2) + x**3*(80*d**4*e**3*g**2 + 320*d**3*e**4*f*g
+ 240*d**2*e**5*f**2) + x**2*(20*d**5*e**2*g**2 + 80*d**4*e**3*f*g + 60*d**3*e**4*f**2) + x*(49*d**6*e*g**2 -
188*d**5*e**2*f*g - 141*d**4*e**3*f**2))/(-480*d**12*e**3 - 1920*d**11*e**4*x - 2400*d**10*e**5*x**2 + 2400*d
**8*e**7*x**4 + 1920*d**7*e**8*x**5 + 480*d**6*e**9*x**6) - (d*g + e*f)*(d*g + 3*e*f)*log(-d*(d*g + e*f)*(d*g
+ 3*e*f)/(e*(d**2*g**2 + 4*d*e*f*g + 3*e**2*f**2)) + x)/(64*d**7*e**3) + (d*g + e*f)*(d*g + 3*e*f)*log(d*(d*g
+ e*f)*(d*g + 3*e*f)/(e*(d**2*g**2 + 4*d*e*f*g + 3*e**2*f**2)) + x)/(64*d**7*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)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError