3.7.59 \(\int \frac {1}{(d f+e f x)^2 (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [659]

Optimal. Leaf size=499 \[ -\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {b \left (5 b^4-47 a b^2 c+124 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {5 b^5-47 a b^3 c+124 a^2 b c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \]

[Out]

-3/8*(-12*a*c+5*b^2)*(-5*a*c+b^2)/a^3/(-4*a*c+b^2)^2/e/f^2/(e*x+d)+1/4*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2
)/e/f^2/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2+1/8*(5*b^4-35*a*b^2*c+36*a^2*c^2+b*c*(-32*a*c+5*b^2)*(e*x+d)^2)/
a^2/(-4*a*c+b^2)^2/e/f^2/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)-3/16*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^
2)^(1/2))^(1/2))*c^(1/2)*((-12*a*c+5*b^2)*(-5*a*c+b^2)+b*(124*a^2*c^2-47*a*b^2*c+5*b^4)/(-4*a*c+b^2)^(1/2))/a^
3/(-4*a*c+b^2)^2/e/f^2*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-3/16*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2
)^(1/2))^(1/2))*c^(1/2)*((-12*a*c+5*b^2)*(-5*a*c+b^2)+(-124*a^2*b*c^2+47*a*b^3*c-5*b^5)/(-4*a*c+b^2)^(1/2))/a^
3/(-4*a*c+b^2)^2/e/f^2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.74, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1156, 1135, 1291, 1295, 1180, 211} \begin {gather*} -\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 e f^2 \left (b^2-4 a c\right )^2 (d+e x)}+\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{8 a^2 e f^2 \left (b^2-4 a c\right )^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 e f^2 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {124 a^2 b c^2-47 a b^3 c+5 b^5}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{8 \sqrt {2} a^3 e f^2 \left (b^2-4 a c\right )^2 \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a e f^2 \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

(-3*(5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(8*a^3*(b^2 - 4*a*c)^2*e*f^2*(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/
(4*a*(b^2 - 4*a*c)*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (5*b^4 - 35*a*b^2*c + 36*a^2*c^2 +
b*c*(5*b^2 - 32*a*c)*(d + e*x)^2)/(8*a^2*(b^2 - 4*a*c)^2*e*f^2*(d + e*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4))
- (3*Sqrt[c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcT
an[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2
- 4*a*c]]*e*f^2) - (3*Sqrt[c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) - (5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2)/Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^2*Sqrt
[b + Sqrt[b^2 - 4*a*c]]*e*f^2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1135

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*
a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*
a*c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7
)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integ
erQ[p] || IntegerQ[m])

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1291

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-(f
*x)^(m + 1))*(a + b*x^2 + c*x^4)^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2)/(2*a*f*(p + 1)*(b^2
- 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[d*(b^2*(m +
2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p]
|| IntegerQ[m])

Rule 1295

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f
*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e f^2}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\text {Subst}\left (\int \frac {-5 b^2+18 a c-7 b c x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{4 a \left (b^2-4 a c\right ) e f^2}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+3 b c \left (5 b^2-32 a c\right ) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{8 a^2 \left (b^2-4 a c\right )^2 e f^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {3 b \left (5 b^4-42 a b^2 c+92 a^2 c^2\right )+3 c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\left (3 c \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{16 a^3 \left (b^2-4 a c\right )^{5/2} e f^2}-\frac {\left (3 c \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {b \left (5 b^4-47 a b^2 c+124 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{16 a^3 \left (b^2-4 a c\right )^2 e f^2}\\ &=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {b \left (5 b^4-47 a b^2 c+124 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e f^2}+\frac {3 \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2-\sqrt {b^2-4 a c} \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^2}\\ \end {align*}

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Mathematica [A]
time = 6.15, size = 575, normalized size = 1.15 \begin {gather*} -\frac {1}{a^3 e f^2 (d+e x)}+\frac {b^3 (d+e x)-3 a b c (d+e x)+b^2 c (d+e x)^3-2 a c^2 (d+e x)^3}{4 a^2 \left (-b^2+4 a c\right ) e f^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {-7 b^5 (d+e x)+52 a b^3 c (d+e x)-84 a^2 b c^2 (d+e x)-7 b^4 c (d+e x)^3+47 a b^2 c^2 (d+e x)^3-52 a^2 c^3 (d+e x)^3}{8 a^3 \left (-b^2+4 a c\right )^2 e f^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {3 \sqrt {c} \left (-5 b^5+47 a b^3 c-124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]

[Out]

-(1/(a^3*e*f^2*(d + e*x))) + (b^3*(d + e*x) - 3*a*b*c*(d + e*x) + b^2*c*(d + e*x)^3 - 2*a*c^2*(d + e*x)^3)/(4*
a^2*(-b^2 + 4*a*c)*e*f^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (-7*b^5*(d + e*x) + 52*a*b^3*c*(d + e*x) - 8
4*a^2*b*c^2*(d + e*x) - 7*b^4*c*(d + e*x)^3 + 47*a*b^2*c^2*(d + e*x)^3 - 52*a^2*c^3*(d + e*x)^3)/(8*a^3*(-b^2
+ 4*a*c)^2*e*f^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4
*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d +
e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^2) - (
3*Sqrt[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a
^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2
- 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e*f^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 1201, normalized size = 2.41

method	result	size
default	\(\frac {-\frac {1}{a^{3} e \left (e x +d \right )}-\frac {\frac {\frac {c^{2} e^{6} \left (52 a^{2} c^{2}-47 a \,b^{2} c +7 b^{4}\right ) x^{7}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {7 c^{2} d \,e^{5} \left (52 a^{2} c^{2}-47 a \,b^{2} c +7 b^{4}\right ) x^{6}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (1092 a^{2} c^{3} d^{2}-987 a \,b^{2} c^{2} d^{2}+147 b^{4} c \,d^{2}+136 a^{2} b \,c^{2}-99 a \,b^{3} c +14 b^{5}\right ) e^{4} c \,x^{5}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {5 c d \,e^{3} \left (364 a^{2} c^{3} d^{2}-329 a \,b^{2} c^{2} d^{2}+49 b^{4} c \,d^{2}+136 a^{2} b \,c^{2}-99 a \,b^{3} c +14 b^{5}\right ) x^{4}}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {e^{2} \left (1820 a^{2} c^{4} d^{4}-1645 a \,b^{2} c^{3} d^{4}+245 b^{4} c^{2} d^{4}+1360 a^{2} b \,c^{3} d^{2}-990 a \,b^{3} c^{2} d^{2}+140 b^{5} c \,d^{2}+68 a^{3} c^{3}+25 a^{2} b^{2} c^{2}-43 a \,b^{4} c +7 b^{6}\right ) x^{3}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {d e \left (1092 a^{2} c^{4} d^{4}-987 a \,b^{2} c^{3} d^{4}+147 b^{4} c^{2} d^{4}+1360 a^{2} b \,c^{3} d^{2}-990 a \,b^{3} c^{2} d^{2}+140 b^{5} c \,d^{2}+204 a^{3} c^{3}+75 a^{2} b^{2} c^{2}-129 a \,b^{4} c +21 b^{6}\right ) x^{2}}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {\left (364 a^{2} c^{4} d^{6}-329 a \,b^{2} c^{3} d^{6}+49 b^{4} c^{2} d^{6}+680 a^{2} b \,c^{3} d^{4}-495 a \,b^{3} c^{2} d^{4}+70 b^{5} c \,d^{4}+204 a^{3} c^{3} d^{2}+75 a^{2} b^{2} c^{2} d^{2}-129 a \,b^{4} c \,d^{2}+21 b^{6} d^{2}+108 a^{3} b \,c^{2}-66 a^{2} b^{3} c +9 a \,b^{5}\right ) x}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}+\frac {d \left (52 a^{2} c^{4} d^{6}-47 a \,b^{2} c^{3} d^{6}+7 b^{4} c^{2} d^{6}+136 a^{2} b \,c^{3} d^{4}-99 a \,b^{3} c^{2} d^{4}+14 b^{5} c \,d^{4}+68 a^{3} c^{3} d^{2}+25 a^{2} b^{2} c^{2} d^{2}-43 a \,b^{4} c \,d^{2}+7 b^{6} d^{2}+108 a^{3} b \,c^{2}-66 a^{2} b^{3} c +9 a \,b^{5}\right )}{8 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (c \,e^{2} \left (60 a^{2} c^{2}-37 a \,b^{2} c +5 b^{4}\right ) \textit {\_R}^{2}+2 c d e \left (60 a^{2} c^{2}-37 a \,b^{2} c +5 b^{4}\right ) \textit {\_R} +60 a^{2} c^{3} d^{2}-37 a \,b^{2} c^{2} d^{2}+5 b^{4} c \,d^{2}+92 a^{2} b \,c^{2}-42 a \,b^{3} c +5 b^{5}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}}{a^{3}}}{f^{2}}\)	\(1201\)
risch	\(\text {Expression too large to display}\)	\(2710\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/f^2*(-1/a^3/e/(e*x+d)-1/a^3*((1/8*c^2*e^6*(52*a^2*c^2-47*a*b^2*c+7*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+7/8*c
^2*d*e^5*(52*a^2*c^2-47*a*b^2*c+7*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/8*(1092*a^2*c^3*d^2-987*a*b^2*c^2*d^2+
147*b^4*c*d^2+136*a^2*b*c^2-99*a*b^3*c+14*b^5)*e^4*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+5/8*c*d*e^3*(364*a^2*c^3*d
^2-329*a*b^2*c^2*d^2+49*b^4*c*d^2+136*a^2*b*c^2-99*a*b^3*c+14*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+1/8*e^2*(182
0*a^2*c^4*d^4-1645*a*b^2*c^3*d^4+245*b^4*c^2*d^4+1360*a^2*b*c^3*d^2-990*a*b^3*c^2*d^2+140*b^5*c*d^2+68*a^3*c^3
+25*a^2*b^2*c^2-43*a*b^4*c+7*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/8*d*e*(1092*a^2*c^4*d^4-987*a*b^2*c^3*d^4+1
47*b^4*c^2*d^4+1360*a^2*b*c^3*d^2-990*a*b^3*c^2*d^2+140*b^5*c*d^2+204*a^3*c^3+75*a^2*b^2*c^2-129*a*b^4*c+21*b^
6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/8*(364*a^2*c^4*d^6-329*a*b^2*c^3*d^6+49*b^4*c^2*d^6+680*a^2*b*c^3*d^4-495*
a*b^3*c^2*d^4+70*b^5*c*d^4+204*a^3*c^3*d^2+75*a^2*b^2*c^2*d^2-129*a*b^4*c*d^2+21*b^6*d^2+108*a^3*b*c^2-66*a^2*
b^3*c+9*a*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/8*d/e*(52*a^2*c^4*d^6-47*a*b^2*c^3*d^6+7*b^4*c^2*d^6+136*a^2*b*c
^3*d^4-99*a*b^3*c^2*d^4+14*b^5*c*d^4+68*a^3*c^3*d^2+25*a^2*b^2*c^2*d^2-43*a*b^4*c*d^2+7*b^6*d^2+108*a^3*b*c^2-
66*a^2*b^3*c+9*a*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*e^4*x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x
^2+c*d^4+2*b*d*e*x+b*d^2+a)^2+3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((c*e^2*(60*a^2*c^2-37*a*b^2*c+5*b^4)*_R^2+
2*c*d*e*(60*a^2*c^2-37*a*b^2*c+5*b^4)*_R+60*a^2*c^3*d^2-37*a*b^2*c^2*d^2+5*b^4*c*d^2+92*a^2*b*c^2-42*a*b^3*c+5
*b^5)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z
^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")

[Out]

-1/8*(3*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^8 + 24*(5*b^4*c^2*e^7 - 37*a*b^2*c^3*e^7 + 60*a^2*c^4*e^7)*d
*x^7 + 3*(5*b^4*c^2*e^8 - 37*a*b^2*c^3*e^8 + 60*a^2*c^4*e^8)*x^8 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*
d^6 + (30*b^5*c*e^6 - 227*a*b^3*c^2*e^6 + 392*a^2*b*c^3*e^6 + 84*(5*b^4*c^2*e^6 - 37*a*b^2*c^3*e^6 + 60*a^2*c^
4*e^6)*d^2)*x^6 + 8*a^2*b^4 - 64*a^3*b^2*c + 128*a^4*c^2 + 6*(28*(5*b^4*c^2*e^5 - 37*a*b^2*c^3*e^5 + 60*a^2*c^
4*e^5)*d^3 + (30*b^5*c*e^5 - 227*a*b^3*c^2*e^5 + 392*a^2*b*c^3*e^5)*d)*x^5 + (15*b^6 - 91*a*b^4*c + 25*a^2*b^2
*c^2 + 324*a^3*c^3)*d^4 + (15*b^6*e^4 - 91*a*b^4*c*e^4 + 25*a^2*b^2*c^2*e^4 + 324*a^3*c^3*e^4 + 210*(5*b^4*c^2
*e^4 - 37*a*b^2*c^3*e^4 + 60*a^2*c^4*e^4)*d^4 + 15*(30*b^5*c*e^4 - 227*a*b^3*c^2*e^4 + 392*a^2*b*c^3*e^4)*d^2)
*x^4 + 4*(42*(5*b^4*c^2*e^3 - 37*a*b^2*c^3*e^3 + 60*a^2*c^4*e^3)*d^5 + 5*(30*b^5*c*e^3 - 227*a*b^3*c^2*e^3 + 3
92*a^2*b*c^3*e^3)*d^3 + (15*b^6*e^3 - 91*a*b^4*c*e^3 + 25*a^2*b^2*c^2*e^3 + 324*a^3*c^3*e^3)*d)*x^3 + (25*a*b^
5 - 194*a^2*b^3*c + 364*a^3*b*c^2)*d^2 + (84*(5*b^4*c^2*e^2 - 37*a*b^2*c^3*e^2 + 60*a^2*c^4*e^2)*d^6 + 25*a*b^
5*e^2 - 194*a^2*b^3*c*e^2 + 364*a^3*b*c^2*e^2 + 15*(30*b^5*c*e^2 - 227*a*b^3*c^2*e^2 + 392*a^2*b*c^3*e^2)*d^4
+ 6*(15*b^6*e^2 - 91*a*b^4*c*e^2 + 25*a^2*b^2*c^2*e^2 + 324*a^3*c^3*e^2)*d^2)*x^2 + 2*(12*(5*b^4*c^2*e - 37*a*
b^2*c^3*e + 60*a^2*c^4*e)*d^7 + 3*(30*b^5*c*e - 227*a*b^3*c^2*e + 392*a^2*b*c^3*e)*d^5 + 2*(15*b^6*e - 91*a*b^
4*c*e + 25*a^2*b^2*c^2*e + 324*a^3*c^3*e)*d^3 + (25*a*b^5*e - 194*a^2*b^3*c*e + 364*a^3*b*c^2*e)*d)*x)/(9*(a^3
*b^4*c^2*e^9 - 8*a^4*b^2*c^3*e^9 + 16*a^5*c^4*e^9)*d*f^2*x^8 + (a^3*b^4*c^2*e^10 - 8*a^4*b^2*c^3*e^10 + 16*a^5
*c^4*e^10)*f^2*x^9 + 2*(a^3*b^5*c*e^8 - 8*a^4*b^3*c^2*e^8 + 16*a^5*b*c^3*e^8 + 18*(a^3*b^4*c^2*e^8 - 8*a^4*b^2
*c^3*e^8 + 16*a^5*c^4*e^8)*d^2)*f^2*x^7 + 14*(6*(a^3*b^4*c^2*e^7 - 8*a^4*b^2*c^3*e^7 + 16*a^5*c^4*e^7)*d^3 + (
a^3*b^5*c*e^7 - 8*a^4*b^3*c^2*e^7 + 16*a^5*b*c^3*e^7)*d)*f^2*x^6 + (a^3*b^6*e^6 - 6*a^4*b^4*c*e^6 + 32*a^6*c^3
*e^6 + 126*(a^3*b^4*c^2*e^6 - 8*a^4*b^2*c^3*e^6 + 16*a^5*c^4*e^6)*d^4 + 42*(a^3*b^5*c*e^6 - 8*a^4*b^3*c^2*e^6
+ 16*a^5*b*c^3*e^6)*d^2)*f^2*x^5 + (126*(a^3*b^4*c^2*e^5 - 8*a^4*b^2*c^3*e^5 + 16*a^5*c^4*e^5)*d^5 + 70*(a^3*b
^5*c*e^5 - 8*a^4*b^3*c^2*e^5 + 16*a^5*b*c^3*e^5)*d^3 + 5*(a^3*b^6*e^5 - 6*a^4*b^4*c*e^5 + 32*a^6*c^3*e^5)*d)*f
^2*x^4 + 2*(a^4*b^5*e^4 - 8*a^5*b^3*c*e^4 + 16*a^6*b*c^2*e^4 + 42*(a^3*b^4*c^2*e^4 - 8*a^4*b^2*c^3*e^4 + 16*a^
5*c^4*e^4)*d^6 + 35*(a^3*b^5*c*e^4 - 8*a^4*b^3*c^2*e^4 + 16*a^5*b*c^3*e^4)*d^4 + 5*(a^3*b^6*e^4 - 6*a^4*b^4*c*
e^4 + 32*a^6*c^3*e^4)*d^2)*f^2*x^3 + 2*(18*(a^3*b^4*c^2*e^3 - 8*a^4*b^2*c^3*e^3 + 16*a^5*c^4*e^3)*d^7 + 21*(a^
3*b^5*c*e^3 - 8*a^4*b^3*c^2*e^3 + 16*a^5*b*c^3*e^3)*d^5 + 5*(a^3*b^6*e^3 - 6*a^4*b^4*c*e^3 + 32*a^6*c^3*e^3)*d
^3 + 3*(a^4*b^5*e^3 - 8*a^5*b^3*c*e^3 + 16*a^6*b*c^2*e^3)*d)*f^2*x^2 + (a^5*b^4*e^2 - 8*a^6*b^2*c*e^2 + 16*a^7
*c^2*e^2 + 9*(a^3*b^4*c^2*e^2 - 8*a^4*b^2*c^3*e^2 + 16*a^5*c^4*e^2)*d^8 + 14*(a^3*b^5*c*e^2 - 8*a^4*b^3*c^2*e^
2 + 16*a^5*b*c^3*e^2)*d^6 + 5*(a^3*b^6*e^2 - 6*a^4*b^4*c*e^2 + 32*a^6*c^3*e^2)*d^4 + 6*(a^4*b^5*e^2 - 8*a^5*b^
3*c*e^2 + 16*a^6*b*c^2*e^2)*d^2)*f^2*x + ((a^3*b^4*c^2*e - 8*a^4*b^2*c^3*e + 16*a^5*c^4*e)*d^9 + 2*(a^3*b^5*c*
e - 8*a^4*b^3*c^2*e + 16*a^5*b*c^3*e)*d^7 + (a^3*b^6*e - 6*a^4*b^4*c*e + 32*a^6*c^3*e)*d^5 + 2*(a^4*b^5*e - 8*
a^5*b^3*c*e + 16*a^6*b*c^2*e)*d^3 + (a^5*b^4*e - 8*a^6*b^2*c*e + 16*a^7*c^2*e)*d)*f^2) - 3/8*integrate((5*b^5
- 42*a*b^3*c + 92*a^2*b*c^2 + (5*b^4*c - 37*a*b^2*c^2 + 60*a^2*c^3)*d^2 + 2*(5*b^4*c*e - 37*a*b^2*c^2*e + 60*a
^2*c^3*e)*d*x + (5*b^4*c*e^2 - 37*a*b^2*c^2*e^2 + 60*a^2*c^3*e^2)*x^2)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*
d^2 + (6*c*d^2*e^2 + b*e^2)*x^2 + 2*(2*c*d^3*e + b*d*e)*x + a), x)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*f^2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10408 vs. \(2 (459) = 918\).
time = 1.17, size = 10408, normalized size = 20.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")

[Out]

-1/16*(6*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*x^8*e^8 + 48*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d*x^7*e^
7 + 6*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^8 + 2*(30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3 + 84*(5*b^4*c^
2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^2)*x^6*e^6 + 2*(30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*d^6 + 12*(28*(5*b^4
*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^3 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*d)*x^5*e^5 + 16*a^2*b^4 - 1
28*a^3*b^2*c + 256*a^4*c^2 + 2*(15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3 + 210*(5*b^4*c^2 - 37*a*b^2
*c^3 + 60*a^2*c^4)*d^4 + 15*(30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*d^2)*x^4*e^4 + 2*(15*b^6 - 91*a*b^4*c +
25*a^2*b^2*c^2 + 324*a^3*c^3)*d^4 + 8*(42*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^5 + 5*(30*b^5*c - 227*a*b
^3*c^2 + 392*a^2*b*c^3)*d^3 + (15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3)*d)*x^3*e^3 + 2*(84*(5*b^4*c
^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^6 + 25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2 + 15*(30*b^5*c - 227*a*b^3*c^2
+ 392*a^2*b*c^3)*d^4 + 6*(15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3)*d^2)*x^2*e^2 + 2*(25*a*b^5 - 194
*a^2*b^3*c + 364*a^3*b*c^2)*d^2 + 4*(12*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^7 + 3*(30*b^5*c - 227*a*b^3*
c^2 + 392*a^2*b*c^3)*d^5 + 2*(15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3)*d^3 + (25*a*b^5 - 194*a^2*b^
3*c + 364*a^3*b*c^2)*d)*x*e - 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*f^2*x^9*e^10 + 9*(a^3*b^
4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d*f^2*x^8*e^9 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3 + 18*(a^3*b^4*
c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^2)*f^2*x^7*e^8 + 14*(6*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^3 + (a
^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d)*f^2*x^6*e^7 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3 + 126*(a^3*b^4*c
^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^4 + 42*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2)*f^2*x^5*e^6 + (126*(
a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5 + 70*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^3 + 5*(a^3*b^6
- 6*a^4*b^4*c + 32*a^6*c^3)*d)*f^2*x^4*e^5 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2 + 42*(a^3*b^4*c^2 - 8*a^
4*b^2*c^3 + 16*a^5*c^4)*d^6 + 35*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^4 + 5*(a^3*b^6 - 6*a^4*b^4*c + 3
2*a^6*c^3)*d^2)*f^2*x^3*e^4 + 2*(18*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^7 + 21*(a^3*b^5*c - 8*a^4*b^3
*c^2 + 16*a^5*b*c^3)*d^5 + 5*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^3 + 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^
2)*d)*f^2*x^2*e^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + 9*(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^8 + 1
4*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^6 + 5*(a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^4 + 6*(a^4*b^5 - 8
*a^5*b^3*c + 16*a^6*b*c^2)*d^2)*f^2*x*e^2 + ((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^9 + 2*(a^3*b^5*c - 8
*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^7 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*d^5 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^
6*b*c^2)*d^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*d)*f^2*e)*sqrt(-(25*b^11 - 495*a*b^9*c + 3894*a^2*b^7*c^2
- 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a
^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*f^4*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a^2*b^8*c^2 - 351
310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/((a^14*b^10 - 20*a^15*b^8*c + 160*a
^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)*f^8)))*e^(-2)/((a^7*b^10 - 20*a^8*b^8*c +
160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*f^4))*log(-27*(4125*b^10*c^4 - 77825*a
*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*x*e - 27*(4125*b^1
0*c^4 - 77825*a*b^8*c^5 + 571030*a^2*b^6*c^6 - 1957349*a^3*b^4*c^7 + 2835000*a^4*b^2*c^8 - 810000*a^5*c^9)*d +
27/2*sqrt(1/2)*((5*a^7*b^16 - 152*a^8*b^14*c + 2006*a^9*b^12*c^2 - 14960*a^10*b^10*c^3 + 68640*a^11*b^8*c^4 -
197120*a^12*b^6*c^5 + 342528*a^13*b^4*c^6 - 323584*a^14*b^2*c^7 + 122880*a^15*c^8)*f^6*sqrt((625*b^12 - 12250
*a*b^10*c + 94725*a^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/
((a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)*f^8))*e
- (125*b^17 - 3775*a*b^15*c + 49360*a^2*b^13*c^2 - 362733*a^3*b^11*c^3 + 1623534*a^4*b^9*c^4 - 4463140*a^5*b^
7*c^5 + 7146736*a^6*b^5*c^6 - 5684672*a^7*b^3*c^7 + 1324800*a^8*b*c^8)*f^2*e)*sqrt(-(25*b^11 - 495*a*b^9*c + 3
894*a^2*b^7*c^2 - 15015*a^3*b^5*c^3 + 27720*a^4*b^3*c^4 - 18480*a^5*b*c^5 + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9
*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*f^4*sqrt((625*b^12 - 12250*a*b^10*c + 94725*a
^2*b^8*c^2 - 351310*a^3*b^6*c^3 + 591886*a^4*b^4*c^4 - 312300*a^5*b^2*c^5 + 50625*a^6*c^6)/((a^14*b^10 - 20*a^
15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)*f^8)))*e^(-2)/((a^7*b^10 -
20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*f^4))) + 3*sqrt(1/2)*(
(a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*f^2*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (459) = 918\).
time = 4.01, size = 1658, normalized size = 3.32 \begin {gather*} -\frac {\frac {7 \, b^{4} c^{2} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f} - \frac {47 \, a b^{2} c^{3} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f} + \frac {52 \, a^{2} c^{4} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f} + \frac {14 \, b^{5} c f e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{3}} - \frac {99 \, a b^{3} c^{2} f e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{3}} + \frac {136 \, a^{2} b c^{3} f e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{3}} + \frac {7 \, b^{6} f^{3} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{5}} - \frac {43 \, a b^{4} c f^{3} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{5}} + \frac {25 \, a^{2} b^{2} c^{2} f^{3} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{5}} + \frac {68 \, a^{3} c^{3} f^{3} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{5}} + \frac {9 \, a b^{5} f^{5} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{7}} - \frac {66 \, a^{2} b^{3} c f^{5} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{7}} + \frac {108 \, a^{3} b c^{2} f^{5} e^{\left (-1\right )}}{{\left (f x e + d f\right )}^{7}}}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} {\left (c + \frac {b f^{2}}{{\left (f x e + d f\right )}^{2}} + \frac {a f^{4}}{{\left (f x e + d f\right )}^{4}}\right )}^{2}} - \frac {e^{\left (-1\right )}}{{\left (f x e + d f\right )} a^{3} f} + \frac {3 \, {\left ({\left (5 \, a^{6} b^{13} - 112 \, a^{7} b^{11} c + 1030 \, a^{8} b^{9} c^{2} - 4928 \, a^{9} b^{7} c^{3} + 12736 \, a^{10} b^{5} c^{4} - 16384 \, a^{11} b^{3} c^{5} + 7680 \, a^{12} b c^{6}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} f^{8} e^{4} + 2 \, {\left (5 \, a^{4} b^{6} c - 57 \, a^{5} b^{4} c^{2} + 208 \, a^{6} b^{2} c^{3} - 240 \, a^{7} c^{4}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} f^{4} {\left | a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2} \right |} e^{2} - {\left (a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2}\right )}^{2} {\left (5 \, b^{5} - 42 \, a b^{3} c + 92 \, a^{2} b c^{2}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f \sqrt {\frac {a^{3} b^{5} f^{4} e^{2} - 8 \, a^{4} b^{3} c f^{4} e^{2} + 16 \, a^{5} b c^{2} f^{4} e^{2} + \sqrt {{\left (a^{3} b^{5} f^{4} e^{2} - 8 \, a^{4} b^{3} c f^{4} e^{2} + 16 \, a^{5} b c^{2} f^{4} e^{2}\right )}^{2} - 4 \, {\left (a^{4} b^{4} f^{8} e^{4} - 8 \, a^{5} b^{2} c f^{8} e^{4} + 16 \, a^{6} c^{2} f^{8} e^{4}\right )} {\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )}}}{a^{4} b^{4} f^{8} e^{4} - 8 \, a^{5} b^{2} c f^{8} e^{4} + 16 \, a^{6} c^{2} f^{8} e^{4}}}}\right ) e^{\left (-3\right )}}{64 \, {\left (a^{7} b^{6} c - 12 \, a^{8} b^{4} c^{2} + 48 \, a^{9} b^{2} c^{3} - 64 \, a^{10} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} f^{6} {\left | a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2} \right |} {\left | a \right |}} - \frac {3 \, {\left ({\left (5 \, a^{6} b^{13} - 112 \, a^{7} b^{11} c + 1030 \, a^{8} b^{9} c^{2} - 4928 \, a^{9} b^{7} c^{3} + 12736 \, a^{10} b^{5} c^{4} - 16384 \, a^{11} b^{3} c^{5} + 7680 \, a^{12} b c^{6}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} f^{8} e^{4} - 2 \, {\left (5 \, a^{4} b^{6} c - 57 \, a^{5} b^{4} c^{2} + 208 \, a^{6} b^{2} c^{3} - 240 \, a^{7} c^{4}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} f^{4} {\left | a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2} \right |} e^{2} - {\left (a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2}\right )}^{2} {\left (5 \, b^{5} - 42 \, a b^{3} c + 92 \, a^{2} b c^{2}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} e^{\left (-1\right )}}{{\left (f x e + d f\right )} f \sqrt {\frac {a^{3} b^{5} f^{4} e^{2} - 8 \, a^{4} b^{3} c f^{4} e^{2} + 16 \, a^{5} b c^{2} f^{4} e^{2} - \sqrt {{\left (a^{3} b^{5} f^{4} e^{2} - 8 \, a^{4} b^{3} c f^{4} e^{2} + 16 \, a^{5} b c^{2} f^{4} e^{2}\right )}^{2} - 4 \, {\left (a^{4} b^{4} f^{8} e^{4} - 8 \, a^{5} b^{2} c f^{8} e^{4} + 16 \, a^{6} c^{2} f^{8} e^{4}\right )} {\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )}}}{a^{4} b^{4} f^{8} e^{4} - 8 \, a^{5} b^{2} c f^{8} e^{4} + 16 \, a^{6} c^{2} f^{8} e^{4}}}}\right ) e^{\left (-3\right )}}{64 \, {\left (a^{7} b^{6} c - 12 \, a^{8} b^{4} c^{2} + 48 \, a^{9} b^{2} c^{3} - 64 \, a^{10} c^{4}\right )} \sqrt {b^{2} - 4 \, a c} f^{6} {\left | a^{3} b^{4} f^{4} e^{2} - 8 \, a^{4} b^{2} c f^{4} e^{2} + 16 \, a^{5} c^{2} f^{4} e^{2} \right |} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*f*x+d*f)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")

[Out]

-1/8*(7*b^4*c^2*e^(-1)/((f*x*e + d*f)*f) - 47*a*b^2*c^3*e^(-1)/((f*x*e + d*f)*f) + 52*a^2*c^4*e^(-1)/((f*x*e +
d*f)*f) + 14*b^5*c*f*e^(-1)/(f*x*e + d*f)^3 - 99*a*b^3*c^2*f*e^(-1)/(f*x*e + d*f)^3 + 136*a^2*b*c^3*f*e^(-1)/
(f*x*e + d*f)^3 + 7*b^6*f^3*e^(-1)/(f*x*e + d*f)^5 - 43*a*b^4*c*f^3*e^(-1)/(f*x*e + d*f)^5 + 25*a^2*b^2*c^2*f^
3*e^(-1)/(f*x*e + d*f)^5 + 68*a^3*c^3*f^3*e^(-1)/(f*x*e + d*f)^5 + 9*a*b^5*f^5*e^(-1)/(f*x*e + d*f)^7 - 66*a^2
*b^3*c*f^5*e^(-1)/(f*x*e + d*f)^7 + 108*a^3*b*c^2*f^5*e^(-1)/(f*x*e + d*f)^7)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5
*c^2)*(c + b*f^2/(f*x*e + d*f)^2 + a*f^4/(f*x*e + d*f)^4)^2) - e^(-1)/((f*x*e + d*f)*a^3*f) + 3/64*((5*a^6*b^1
3 - 112*a^7*b^11*c + 1030*a^8*b^9*c^2 - 4928*a^9*b^7*c^3 + 12736*a^10*b^5*c^4 - 16384*a^11*b^3*c^5 + 7680*a^12
*b*c^6)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*f^8*e^4 + 2*(5*a^4*b^6*c - 57*a^5*b^4*c^2 + 208*a^6*b^2*c^3 - 240*
a^7*c^4)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*f^4*abs(a^3*b^4*f^4*e^2 - 8*a^4*b^2*c*f^4*e^2 +
16*a^5*c^2*f^4*e^2)*e^2 - (a^3*b^4*f^4*e^2 - 8*a^4*b^2*c*f^4*e^2 + 16*a^5*c^2*f^4*e^2)^2*(5*b^5 - 42*a*b^3*c
+ 92*a^2*b*c^2)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)*e^(-1)/((f*x*e + d*f)*f*sqrt((a^3*b^5*
f^4*e^2 - 8*a^4*b^3*c*f^4*e^2 + 16*a^5*b*c^2*f^4*e^2 + sqrt((a^3*b^5*f^4*e^2 - 8*a^4*b^3*c*f^4*e^2 + 16*a^5*b*
c^2*f^4*e^2)^2 - 4*(a^4*b^4*f^8*e^4 - 8*a^5*b^2*c*f^8*e^4 + 16*a^6*c^2*f^8*e^4)*(a^3*b^4*c - 8*a^4*b^2*c^2 + 1
6*a^5*c^3)))/(a^4*b^4*f^8*e^4 - 8*a^5*b^2*c*f^8*e^4 + 16*a^6*c^2*f^8*e^4))))*e^(-3)/((a^7*b^6*c - 12*a^8*b^4*c
^2 + 48*a^9*b^2*c^3 - 64*a^10*c^4)*sqrt(b^2 - 4*a*c)*f^6*abs(a^3*b^4*f^4*e^2 - 8*a^4*b^2*c*f^4*e^2 + 16*a^5*c^
2*f^4*e^2)*abs(a)) - 3/64*((5*a^6*b^13 - 112*a^7*b^11*c + 1030*a^8*b^9*c^2 - 4928*a^9*b^7*c^3 + 12736*a^10*b^5
*c^4 - 16384*a^11*b^3*c^5 + 7680*a^12*b*c^6)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*f^8*e^4 - 2*(5*a^4*b^6*c - 57
*a^5*b^4*c^2 + 208*a^6*b^2*c^3 - 240*a^7*c^4)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*f^4*abs(a^
3*b^4*f^4*e^2 - 8*a^4*b^2*c*f^4*e^2 + 16*a^5*c^2*f^4*e^2)*e^2 - (a^3*b^4*f^4*e^2 - 8*a^4*b^2*c*f^4*e^2 + 16*a^
5*c^2*f^4*e^2)^2*(5*b^5 - 42*a*b^3*c + 92*a^2*b*c^2)*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)*e
^(-1)/((f*x*e + d*f)*f*sqrt((a^3*b^5*f^4*e^2 - 8*a^4*b^3*c*f^4*e^2 + 16*a^5*b*c^2*f^4*e^2 - sqrt((a^3*b^5*f^4*
e^2 - 8*a^4*b^3*c*f^4*e^2 + 16*a^5*b*c^2*f^4*e^2)^2 - 4*(a^4*b^4*f^8*e^4 - 8*a^5*b^2*c*f^8*e^4 + 16*a^6*c^2*f^
8*e^4)*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)))/(a^4*b^4*f^8*e^4 - 8*a^5*b^2*c*f^8*e^4 + 16*a^6*c^2*f^8*e^4)
)))*e^(-3)/((a^7*b^6*c - 12*a^8*b^4*c^2 + 48*a^9*b^2*c^3 - 64*a^10*c^4)*sqrt(b^2 - 4*a*c)*f^6*abs(a^3*b^4*f^4*
e^2 - 8*a^4*b^2*c*f^4*e^2 + 16*a^5*c^2*f^4*e^2)*abs(a))

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Mupad [B]
time = 15.40, size = 2500, normalized size = 5.01 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*f + e*f*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x)

[Out]

- atan(((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*
a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 626
84160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*
c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^7*b^20*e^2*f^4 + 1048576*a^17*
c^10*e^2*f^4 + 720*a^9*b^16*c^2*e^2*f^4 - 7680*a^10*b^14*c^3*e^2*f^4 + 53760*a^11*b^12*c^4*e^2*f^4 - 258048*a^
12*b^10*c^5*e^2*f^4 + 860160*a^13*b^8*c^6*e^2*f^4 - 1966080*a^14*b^6*c^7*e^2*f^4 + 2949120*a^15*b^4*c^8*e^2*f^
4 - 2621440*a^16*b^2*c^9*e^2*f^4 - 40*a^8*b^18*c*e^2*f^4)))^(1/2)*(x*(271790899200*a^20*c^14*e^12*f^6 - 230400
*a^9*b^22*c^3*e^12*f^6 + 9861120*a^10*b^20*c^4*e^12*f^6 - 191038464*a^11*b^18*c^5*e^12*f^6 + 2207803392*a^12*b
^16*c^6*e^12*f^6 - 16878108672*a^13*b^14*c^7*e^12*f^6 + 89374851072*a^14*b^12*c^8*e^12*f^6 - 333226967040*a^15
*b^10*c^9*e^12*f^6 + 869815812096*a^16*b^8*c^10*e^12*f^6 - 1543847804928*a^17*b^6*c^11*e^12*f^6 + 174731349196
8*a^18*b^4*c^12*e^12*f^6 - 1101055131648*a^19*b^2*c^13*e^12*f^6) - (-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^
(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b
^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*
c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*
c - b^2)^15)^(1/2)))/(512*(a^7*b^20*e^2*f^4 + 1048576*a^17*c^10*e^2*f^4 + 720*a^9*b^16*c^2*e^2*f^4 - 7680*a^10
*b^14*c^3*e^2*f^4 + 53760*a^11*b^12*c^4*e^2*f^4 - 258048*a^12*b^10*c^5*e^2*f^4 + 860160*a^13*b^8*c^6*e^2*f^4 -
1966080*a^14*b^6*c^7*e^2*f^4 + 2949120*a^15*b^4*c^8*e^2*f^4 - 2621440*a^16*b^2*c^9*e^2*f^4 - 40*a^8*b^18*c*e^
2*f^4)))^(1/2)*((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^2 -
188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c
^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - 694*
a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^7*b^20*e^2*f^4 + 10485
76*a^17*c^10*e^2*f^4 + 720*a^9*b^16*c^2*e^2*f^4 - 7680*a^10*b^14*c^3*e^2*f^4 + 53760*a^11*b^12*c^4*e^2*f^4 - 2
58048*a^12*b^10*c^5*e^2*f^4 + 860160*a^13*b^8*c^6*e^2*f^4 - 1966080*a^14*b^6*c^7*e^2*f^4 + 2949120*a^15*b^4*c^
8*e^2*f^4 - 2621440*a^16*b^2*c^9*e^2*f^4 - 40*a^8*b^18*c*e^2*f^4)))^(1/2)*(x*(262144*a^15*b^23*c^2*e^14*f^10 -
11534336*a^16*b^21*c^3*e^14*f^10 + 230686720*a^17*b^19*c^4*e^14*f^10 - 2768240640*a^18*b^17*c^5*e^14*f^10 + 2
2145925120*a^19*b^15*c^6*e^14*f^10 - 124017180672*a^20*b^13*c^7*e^14*f^10 + 496068722688*a^21*b^11*c^8*e^14*f^
10 - 1417339207680*a^22*b^9*c^9*e^14*f^10 + 2834678415360*a^23*b^7*c^10*e^14*f^10 - 3779571220480*a^24*b^5*c^1
1*e^14*f^10 + 3023656976384*a^25*b^3*c^12*e^14*f^10 - 1099511627776*a^26*b*c^13*e^14*f^10) - 1099511627776*a^2
6*b*c^13*d*e^13*f^10 + 262144*a^15*b^23*c^2*d*e^13*f^10 - 11534336*a^16*b^21*c^3*d*e^13*f^10 + 230686720*a^17*
b^19*c^4*d*e^13*f^10 - 2768240640*a^18*b^17*c^5*d*e^13*f^10 + 22145925120*a^19*b^15*c^6*d*e^13*f^10 - 12401718
0672*a^20*b^13*c^7*d*e^13*f^10 + 496068722688*a^21*b^11*c^8*d*e^13*f^10 - 1417339207680*a^22*b^9*c^9*d*e^13*f^
10 + 2834678415360*a^23*b^7*c^10*d*e^13*f^10 - 3779571220480*a^24*b^5*c^11*d*e^13*f^10 + 3023656976384*a^25*b^
3*c^12*d*e^13*f^10) - 245760*a^12*b^23*c^2*e^12*f^8 + 10911744*a^13*b^21*c^3*e^12*f^8 - 220397568*a^14*b^19*c^
4*e^12*f^8 + 2673082368*a^15*b^17*c^5*e^12*f^8 - 21630025728*a^16*b^15*c^6*e^12*f^8 + 122607894528*a^17*b^13*c
^7*e^12*f^8 - 496773365760*a^18*b^11*c^8*e^12*f^8 + 1438679826432*a^19*b^9*c^9*e^12*f^8 - 2918430277632*a^20*b
^7*c^10*e^12*f^8 + 3949222428672*a^21*b^5*c^11*e^12*f^8 - 3208340570112*a^22*b^3*c^12*e^12*f^8 + 1185410973696
*a^23*b*c^13*e^12*f^8) + 271790899200*a^20*c^14*d*e^11*f^6 - 230400*a^9*b^22*c^3*d*e^11*f^6 + 9861120*a^10*b^2
0*c^4*d*e^11*f^6 - 191038464*a^11*b^18*c^5*d*e^11*f^6 + 2207803392*a^12*b^16*c^6*d*e^11*f^6 - 16878108672*a^13
*b^14*c^7*d*e^11*f^6 + 89374851072*a^14*b^12*c^8*d*e^11*f^6 - 333226967040*a^15*b^10*c^9*d*e^11*f^6 + 86981581
2096*a^16*b^8*c^10*d*e^11*f^6 - 1543847804928*a^17*b^6*c^11*d*e^11*f^6 + 1747313491968*a^18*b^4*c^12*d*e^11*f^
6 - 1101055131648*a^19*b^2*c^13*d*e^11*f^6)*1i + (-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a
^10*b*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19905600
*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2
)^15)^(1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2)
))/(512*(a^7*b^20*e^2*f^4 + 1048576*a^17*c^10*e^2*f^4 + 720*a^9*b^16*c^2*e^2*f^4 - 7680*a^10*b^14*c^3*e^2*f^4
+ 53760*a^11*b^12*c^4*e^2*f^4 - 258048*a^12*b^1...

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