3.7.53 \(\int \frac {1}{(d f+e f x)^4 (a+b (d+e x)^2+c (d+e x)^4)^2} \, dx\) [653]
Optimal. Leaf size=423 \[ -\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e f^4 (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e f^4 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e f^4}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^4} \]
[Out]
1/6*(14*a*c-5*b^2)/a^2/(-4*a*c+b^2)/e/f^4/(e*x+d)^3+1/2*b*(-19*a*c+5*b^2)/a^3/(-4*a*c+b^2)/e/f^4/(e*x+d)+1/2*(
b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f^4/(e*x+d)^3/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/4*arctan((e*x+d)*2^(1/2)
*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/
2))/a^3/(-4*a*c+b^2)^(3/2)/e/f^4*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/4*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b+(-
4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(5*b^4-29*a*b^2*c+28*a^2*c^2-b*(-19*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))/a^3/(-4*a*
c+b^2)^(3/2)/e/f^4*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
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Rubi [A]
time = 2.38, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1156, 1135,
1295, 1180, 211} \begin {gather*} \frac {b \left (5 b^2-19 a c\right )}{2 a^3 e f^4 \left (b^2-4 a c\right ) (d+e x)}-\frac {5 b^2-14 a c}{6 a^2 e f^4 \left (b^2-4 a c\right ) (d+e x)^3}+\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^3 e f^4 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 \sqrt {2} a^3 e f^4 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a e f^4 \left (b^2-4 a c\right ) (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d*f + e*f*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]
[Out]
-1/6*(5*b^2 - 14*a*c)/(a^2*(b^2 - 4*a*c)*e*f^4*(d + e*x)^3) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*e*f^4*
(d + e*x)) + (b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*e*f^4*(d + e*x)^3*(a + b*(d + e*x)^2 + c*(d +
e*x)^4)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*S
qrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]
*e*f^4) - (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*
e*f^4)
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 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)^4 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e f^4}\\ &=\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-5 b^2+14 a c-5 b c x^2}{x^4 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{2 a \left (b^2-4 a c\right ) e f^4}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e f^4 (d+e x)^3}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\text {Subst}\left (\int \frac {-3 b \left (5 b^2-19 a c\right )-3 c \left (5 b^2-14 a c\right ) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{6 a^2 \left (b^2-4 a c\right ) e f^4}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e f^4 (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e f^4 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 b^4-24 a b^2 c+14 a^2 c^2\right )-3 b c \left (5 b^2-19 a c\right ) x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{6 a^3 \left (b^2-4 a c\right ) e f^4}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e f^4 (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e f^4 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\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 )}{4 a^3 \left (b^2-4 a c\right )^{3/2} e f^4}+\frac {\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\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 )}{4 a^3 \left (b^2-4 a c\right )^{3/2} e f^4}\\ &=-\frac {5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) e f^4 (d+e x)^3}+\frac {b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) e f^4 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) e f^4 (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 a c\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 )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e f^4}-\frac {\sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 a c\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 )}{2 \sqrt {2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^4}\\ \end {align*}
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Mathematica [A]
time = 1.91, size = 387, normalized size = 0.91 \begin {gather*} \frac {-\frac {4 a}{(d+e x)^3}+\frac {24 b}{d+e x}+\frac {6 (d+e x) \left (b^4-4 a b^2 c+2 a^2 c^2+b^3 c (d+e x)^2-3 a b c^2 (d+e x)^2\right )}{\left (b^2-4 a c\right ) \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \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 )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-5 b^4+29 a b^2 c-28 a^2 c^2+5 b^3 \sqrt {b^2-4 a c}-19 a b c \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 )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{12 a^3 e f^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d*f + e*f*x)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x]
[Out]
((-4*a)/(d + e*x)^3 + (24*b)/(d + e*x) + (6*(d + e*x)*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*(d + e*x)^2 - 3*a*b
*c^2*(d + e*x)^2))/((b^2 - 4*a*c)*(a + (d + e*x)^2*(b + c*(d + e*x)^2))) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^
2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sq
rt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 2
9*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*
x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(12*a^3*e*f^4)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 493, normalized size = 1.17
| | |
| method |
result |
size |
| | |
| default |
\(\frac {-\frac {1}{3 a^{2} e \left (e x +d \right )^{3}}+\frac {2 b}{a^{3} e \left (e x +d \right )}-\frac {\frac {-\frac {b c \,e^{2} \left (3 a c -b^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right )}-\frac {3 d b c e \left (3 a c -b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (-9 a b \,c^{2} d^{2}+3 b^{3} c \,d^{2}+2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{8 a c -2 b^{2}}+\frac {d \left (-3 a b \,c^{2} d^{2}+b^{3} c \,d^{2}+2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right )}{2 e \left (4 a c -b^{2}\right )}}{c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (b c \,e^{2} \left (-19 a c +5 b^{2}\right ) \textit {\_R}^{2}+2 b c d e \left (-19 a c +5 b^{2}\right ) \textit {\_R} -19 a b \,c^{2} d^{2}+5 b^{3} c \,d^{2}+14 a^{2} c^{2}-24 a \,b^{2} c +5 b^{4}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{4 \left (4 a c -b^{2}\right ) e}}{a^{3}}}{f^{4}}\) |
\(493\) |
| risch |
\(\text {Expression too large to display}\) |
\(1565\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*f*x+d*f)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x,method=_RETURNVERBOSE)
[Out]
1/f^4*(-1/3/a^2/e/(e*x+d)^3+2/a^3*b/e/(e*x+d)-1/a^3*((-1/2*b*c*e^2*(3*a*c-b^2)/(4*a*c-b^2)*x^3-3/2*d*b*c*e*(3*
a*c-b^2)/(4*a*c-b^2)*x^2+1/2*(-9*a*b*c^2*d^2+3*b^3*c*d^2+2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2)*x+1/2*d/e*(-3*a*
b*c^2*d^2+b^3*c*d^2+2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2))/(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)+1/4/(4*a*c-b^2)/e*sum((b*c*e^2*(-19*a*c+5*b^2)*_R^2+2*b*c*d*e*(-19*a*c+5*b
^2)*_R-19*a*b*c^2*d^2+5*b^3*c*d^2+14*a^2*c^2-24*a*b^2*c+5*b^4)/(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)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="maxima")
[Out]
1/6*(3*(5*b^3*c - 19*a*b*c^2)*d^6 + 18*(5*b^3*c*e^5 - 19*a*b*c^2*e^5)*d*x^5 + 3*(5*b^3*c*e^6 - 19*a*b*c^2*e^6)
*x^6 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d^4 + (15*b^4*e^4 - 62*a*b^2*c*e^4 + 14*a^2*c^2*e^4 + 45*(5*b^3*c*e^
4 - 19*a*b*c^2*e^4)*d^2)*x^4 - 2*a^2*b^2 + 8*a^3*c + 4*(15*(5*b^3*c*e^3 - 19*a*b*c^2*e^3)*d^3 + (15*b^4*e^3 -
62*a*b^2*c*e^3 + 14*a^2*c^2*e^3)*d)*x^3 + 10*(a*b^3 - 4*a^2*b*c)*d^2 + (45*(5*b^3*c*e^2 - 19*a*b*c^2*e^2)*d^4
+ 10*a*b^3*e^2 - 40*a^2*b*c*e^2 + 6*(15*b^4*e^2 - 62*a*b^2*c*e^2 + 14*a^2*c^2*e^2)*d^2)*x^2 + 2*(9*(5*b^3*c*e
- 19*a*b*c^2*e)*d^5 + 2*(15*b^4*e - 62*a*b^2*c*e + 14*a^2*c^2*e)*d^3 + 10*(a*b^3*e - 4*a^2*b*c*e)*d)*x)/(7*(a^
3*b^2*c*e^7 - 4*a^4*c^2*e^7)*d*f^4*x^6 + (a^3*b^2*c*e^8 - 4*a^4*c^2*e^8)*f^4*x^7 + (a^3*b^3*e^6 - 4*a^4*b*c*e^
6 + 21*(a^3*b^2*c*e^6 - 4*a^4*c^2*e^6)*d^2)*f^4*x^5 + 5*(7*(a^3*b^2*c*e^5 - 4*a^4*c^2*e^5)*d^3 + (a^3*b^3*e^5
- 4*a^4*b*c*e^5)*d)*f^4*x^4 + (a^4*b^2*e^4 - 4*a^5*c*e^4 + 35*(a^3*b^2*c*e^4 - 4*a^4*c^2*e^4)*d^4 + 10*(a^3*b^
3*e^4 - 4*a^4*b*c*e^4)*d^2)*f^4*x^3 + (21*(a^3*b^2*c*e^3 - 4*a^4*c^2*e^3)*d^5 + 10*(a^3*b^3*e^3 - 4*a^4*b*c*e^
3)*d^3 + 3*(a^4*b^2*e^3 - 4*a^5*c*e^3)*d)*f^4*x^2 + (7*(a^3*b^2*c*e^2 - 4*a^4*c^2*e^2)*d^6 + 5*(a^3*b^3*e^2 -
4*a^4*b*c*e^2)*d^4 + 3*(a^4*b^2*e^2 - 4*a^5*c*e^2)*d^2)*f^4*x + ((a^3*b^2*c*e - 4*a^4*c^2*e)*d^7 + (a^3*b^3*e
- 4*a^4*b*c*e)*d^5 + (a^4*b^2*e - 4*a^5*c*e)*d^3)*f^4) + 1/2*integrate((5*b^4 - 24*a*b^2*c + 14*a^2*c^2 + (5*b
^3*c - 19*a*b*c^2)*d^2 + 2*(5*b^3*c*e - 19*a*b*c^2*e)*d*x + (5*b^3*c*e^2 - 19*a*b*c^2*e^2)*x^2)/((b^2*c - 4*a*
c^2)*d^4 + 4*(b^2*c*e^3 - 4*a*c^2*e^3)*d*x^3 + (b^2*c*e^4 - 4*a*c^2*e^4)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*
c)*d^2 + (b^3*e^2 - 4*a*b*c*e^2 + 6*(b^2*c*e^2 - 4*a*c^2*e^2)*d^2)*x^2 + 2*(2*(b^2*c*e - 4*a*c^2*e)*d^3 + (b^3
*e - 4*a*b*c*e)*d)*x), x)/(a^3*f^4)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5856 vs.
\(2 (376) = 752\).
time = 0.72, size = 5856, normalized size = 13.84 \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)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="fricas")
[Out]
1/12*(6*(5*b^3*c - 19*a*b*c^2)*x^6*e^6 + 36*(5*b^3*c - 19*a*b*c^2)*d*x^5*e^5 + 6*(5*b^3*c - 19*a*b*c^2)*d^6 +
2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2 + 45*(5*b^3*c - 19*a*b*c^2)*d^2)*x^4*e^4 + 2*(15*b^4 - 62*a*b^2*c + 14*a^2
*c^2)*d^4 + 8*(15*(5*b^3*c - 19*a*b*c^2)*d^3 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d)*x^3*e^3 - 4*a^2*b^2 + 16*
a^3*c + 2*(45*(5*b^3*c - 19*a*b*c^2)*d^4 + 10*a*b^3 - 40*a^2*b*c + 6*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d^2)*x
^2*e^2 + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*f^4*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c^2)*d*f^4*x^6*e^7 + (a^3*b^3
- 4*a^4*b*c + 21*(a^3*b^2*c - 4*a^4*c^2)*d^2)*f^4*x^5*e^6 + 5*(7*(a^3*b^2*c - 4*a^4*c^2)*d^3 + (a^3*b^3 - 4*a
^4*b*c)*d)*f^4*x^4*e^5 + (a^4*b^2 - 4*a^5*c + 35*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 10*(a^3*b^3 - 4*a^4*b*c)*d^2)*f
^4*x^3*e^4 + (21*(a^3*b^2*c - 4*a^4*c^2)*d^5 + 10*(a^3*b^3 - 4*a^4*b*c)*d^3 + 3*(a^4*b^2 - 4*a^5*c)*d)*f^4*x^2
*e^3 + (7*(a^3*b^2*c - 4*a^4*c^2)*d^6 + 5*(a^3*b^3 - 4*a^4*b*c)*d^4 + 3*(a^4*b^2 - 4*a^5*c)*d^2)*f^4*x*e^2 + (
(a^3*b^2*c - 4*a^4*c^2)*d^7 + (a^3*b^3 - 4*a^4*b*c)*d^5 + (a^4*b^2 - 4*a^5*c)*d^3)*f^4*e)*sqrt(((a^7*b^6 - 12*
a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b
^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 -
64*a^17*c^3)*f^16)) - 25*b^9 + 315*a*b^7*c - 1386*a^2*b^5*c^2 + 2415*a^3*b^3*c^3 - 1260*a^4*b*c^4)/((a^7*b^6 -
12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8))*e^(-1)*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4
*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*x*e + 1/2*sqrt(1/2)*((5*a^7*b^11 - 94*a^8*b^9*c + 700*a^9*b^7*c^2 - 2
576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*f^12*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^
2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^14*b^6 - 12*a^15*b^4*c + 48*
a^16*b^2*c^2 - 64*a^17*c^3)*f^16))*e + (125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 16
0932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7)*f^4*e)*sqrt(((a^7*b^6 - 12*a^8*b^4*
c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 +
76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*
c^3)*f^16)) - 25*b^9 + 315*a*b^7*c - 1386*a^2*b^5*c^2 + 2415*a^3*b^3*c^3 - 1260*a^4*b*c^4)/((a^7*b^6 - 12*a^8*
b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8))*e^(-1) + (1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 504
21*a^3*b^2*c^7 + 9604*a^4*c^8)*d) - 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*f^4*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c^
2)*d*f^4*x^6*e^7 + (a^3*b^3 - 4*a^4*b*c + 21*(a^3*b^2*c - 4*a^4*c^2)*d^2)*f^4*x^5*e^6 + 5*(7*(a^3*b^2*c - 4*a^
4*c^2)*d^3 + (a^3*b^3 - 4*a^4*b*c)*d)*f^4*x^4*e^5 + (a^4*b^2 - 4*a^5*c + 35*(a^3*b^2*c - 4*a^4*c^2)*d^4 + 10*(
a^3*b^3 - 4*a^4*b*c)*d^2)*f^4*x^3*e^4 + (21*(a^3*b^2*c - 4*a^4*c^2)*d^5 + 10*(a^3*b^3 - 4*a^4*b*c)*d^3 + 3*(a^
4*b^2 - 4*a^5*c)*d)*f^4*x^2*e^3 + (7*(a^3*b^2*c - 4*a^4*c^2)*d^6 + 5*(a^3*b^3 - 4*a^4*b*c)*d^4 + 3*(a^4*b^2 -
4*a^5*c)*d^2)*f^4*x*e^2 + ((a^3*b^2*c - 4*a^4*c^2)*d^7 + (a^3*b^3 - 4*a^4*b*c)*d^5 + (a^4*b^2 - 4*a^5*c)*d^3)*
f^4*e)*sqrt(((a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8*sqrt((625*b^12 - 8250*a*b^10*c + 3952
5*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^14*b^6 - 12*a^15
*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)*f^16)) - 25*b^9 + 315*a*b^7*c - 1386*a^2*b^5*c^2 + 2415*a^3*b^3*c^3 -
1260*a^4*b*c^4)/((a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8))*e^(-1)*log((1125*b^8*c^4 - 1232
5*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*x*e - 1/2*sqrt(1/2)*((5*a^7*b^11 - 94*a^8*
b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*f^12*sqrt((625*b^12 - 8250*
a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^1
4*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)*f^16))*e + (125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c
^2 - 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7)*f^4*e)*s
qrt(((a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^
8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/((a^14*b^6 - 12*a^15*b^4*c +
48*a^16*b^2*c^2 - 64*a^17*c^3)*f^16)) - 25*b^9 + 315*a*b^7*c - 1386*a^2*b^5*c^2 + 2415*a^3*b^3*c^3 - 1260*a^4
*b*c^4)/((a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*f^8))*e^(-1) + (1125*b^8*c^4 - 12325*a*b^6*c^
5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*d) + 20*(a*b^3 - 4*a^2*b*c)*d^2 + 4*(9*(5*b^3*c - 19
*a*b*c^2)*d^5 + 2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*d^3 + 10*(a*b^3 - 4*a^2*b*c)*d)*x*e - 3*sqrt(1/2)*((a^3*b
^2*c - 4*a^4*c^2)*f^4*x^7*e^8 + 7*(a^3*b^2*c - 4*a^4*c^2)*d*f^4*x^6*e^7 + (a^3*b^3 - 4*a^4*b*c + 21*(a^3*b^2*c
- 4*a^4*c^2)*d^2)*f^4*x^5*e^6 + 5*(7*(a^3*b^2*...
________________________________________________________________________________________
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)**4/(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2002 vs.
\(2 (376) = 752\).
time = 3.22, size = 2002, normalized size = 4.73 \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)^4/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="giac")
[Out]
-1/4*((5*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 19*(d*e^(-1) + s
qrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^
2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^
2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*log(d*e^(-1) + x +
sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c
*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)
*e^(-4)/c))) + (5*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2 - 19*(d*e
^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) - sqrt(1/2)*sq
rt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*log(d*e^(
-1) + x - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + s
qrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4
)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*
a*c)*e^2)*e^(-4)/c))) + (5*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b^3*c*e^2
- 19*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(-1) + sqr
t(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sq
rt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*a^2*c^2)*
log(d*e^(-1) + x + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(
b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e
^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt
(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + (5*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b
^3*c*e^2 - 19*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*a*b*c^2*e^2 - 10*(d*e^(
-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b^3*c*d*e + 38*(d*e^(-1) - sqrt(1/2)*sqrt(-(b
*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*a*b*c^2*d*e + 5*b^3*c*d^2 - 19*a*b*c^2*d^2 + 5*b^4 - 24*a*b^2*c + 14*
a^2*c^2)*log(d*e^(-1) + x - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2
)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 -
4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e
^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))))/(a^3*b^2*f^4 - 4*a^4*c*f^4) + 1/2*(b^3*c*x^3*e^3 - 3*a*b*c^2*x^3*e^3
+ 3*b^3*c*d*x^2*e^2 - 9*a*b*c^2*d*x^2*e^2 + 3*b^3*c*d^2*x*e - 9*a*b*c^2*d^2*x*e + b^3*c*d^3 - 3*a*b*c^2*d^3 +
b^4*x*e - 4*a*b^2*c*x*e + 2*a^2*c^2*x*e + b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d)/((a^3*b^2*f^4*e - 4*a^4*c*f^4*e)*
(c*x^4*e^4 + 4*c*d*x^3*e^3 + 6*c*d^2*x^2*e^2 + 4*c*d^3*x*e + c*d^4 + b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a)) + 1/3
*(6*b*x^2*e^2 + 12*b*d*x*e + 6*b*d^2 - a)*e^(-1)/((x*e + d)^3*a^3*f^4)
________________________________________________________________________________________
Mupad [B]
time = 10.45, size = 2500, normalized size = 5.91 \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)^4*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2),x)
[Out]
atan(((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 +
116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^
13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2*f^8
+ 4096*a^13*c^6*e^2*f^8 + 240*a^9*b^8*c^2*e^2*f^8 - 1280*a^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^4*c^4*e^2*f^8 - 61
44*a^12*b^2*c^5*e^2*f^8 - 24*a^8*b^10*c*e^2*f^8)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640
*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*
c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^
4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2*f^8 + 4096*a^13*c^6*e^2*f^8 + 240*a^9*b^8*c^2*e^2*f^8 - 1280*a
^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^4*c^4*e^2*f^8 - 6144*a^12*b^2*c^5*e^2*f^8 - 24*a^8*b^10*c*e^2*f^8)))^(1/2)*(
(-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 11692
8*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c -
246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2*f^8 + 4096
*a^13*c^6*e^2*f^8 + 240*a^9*b^8*c^2*e^2*f^8 - 1280*a^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^4*c^4*e^2*f^8 - 6144*a^1
2*b^2*c^5*e^2*f^8 - 24*a^8*b^10*c*e^2*f^8)))^(1/2)*(x*(256*a^15*b^13*c^2*e^14*f^20 - 6144*a^16*b^11*c^3*e^14*f
^20 + 61440*a^17*b^9*c^4*e^14*f^20 - 327680*a^18*b^7*c^5*e^14*f^20 + 983040*a^19*b^5*c^6*e^14*f^20 - 1572864*a
^20*b^3*c^7*e^14*f^20 + 1048576*a^21*b*c^8*e^14*f^20) + 1048576*a^21*b*c^8*d*e^13*f^20 + 256*a^15*b^13*c^2*d*e
^13*f^20 - 6144*a^16*b^11*c^3*d*e^13*f^20 + 61440*a^17*b^9*c^4*d*e^13*f^20 - 327680*a^18*b^7*c^5*d*e^13*f^20 +
983040*a^19*b^5*c^6*d*e^13*f^20 - 1572864*a^20*b^3*c^7*d*e^13*f^20) - 917504*a^19*c^9*e^12*f^16 + 320*a^12*b^
14*c^2*e^12*f^16 - 7936*a^13*b^12*c^3*e^12*f^16 + 82816*a^14*b^10*c^4*e^12*f^16 - 468480*a^15*b^8*c^5*e^12*f^1
6 + 1536000*a^16*b^6*c^6*e^12*f^16 - 2867200*a^17*b^4*c^7*e^12*f^16 + 2719744*a^18*b^2*c^8*e^12*f^16) - x*(401
408*a^16*c^10*e^12*f^12 - 400*a^9*b^14*c^3*e^12*f^12 + 9440*a^10*b^12*c^4*e^12*f^12 - 92816*a^11*b^10*c^5*e^12
*f^12 + 488096*a^12*b^8*c^6*e^12*f^12 - 1458688*a^13*b^6*c^7*e^12*f^12 + 2401280*a^14*b^4*c^8*e^12*f^12 - 1871
872*a^15*b^2*c^9*e^12*f^12) - 401408*a^16*c^10*d*e^11*f^12 + 400*a^9*b^14*c^3*d*e^11*f^12 - 9440*a^10*b^12*c^4
*d*e^11*f^12 + 92816*a^11*b^10*c^5*d*e^11*f^12 - 488096*a^12*b^8*c^6*d*e^11*f^12 + 1458688*a^13*b^6*c^7*d*e^11
*f^12 - 2401280*a^14*b^4*c^8*d*e^11*f^12 + 1871872*a^15*b^2*c^9*d*e^11*f^12)*1i + (-(25*b^15 - 25*b^6*(-(4*a*c
- b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b
^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c -
b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2*f^8 + 4096*a^13*c^6*e^2*f^8 + 240*a^9*
b^8*c^2*e^2*f^8 - 1280*a^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^4*c^4*e^2*f^8 - 6144*a^12*b^2*c^5*e^2*f^8 - 24*a^8*b
^10*c*e^2*f^8)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 3
5767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5 + 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9
)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*
(a^7*b^12*e^2*f^8 + 4096*a^13*c^6*e^2*f^8 + 240*a^9*b^8*c^2*e^2*f^8 - 1280*a^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^
4*c^4*e^2*f^8 - 6144*a^12*b^2*c^5*e^2*f^8 - 24*a^8*b^10*c*e^2*f^8)))^(1/2)*((-(25*b^15 - 25*b^6*(-(4*a*c - b^2
)^9)^(1/2) - 80640*a^7*b*c^7 + 6366*a^2*b^11*c^2 - 35767*a^3*b^9*c^3 + 116928*a^4*b^7*c^4 - 219744*a^5*b^5*c^5
+ 215040*a^6*b^3*c^6 + 49*a^3*c^3*(-(4*a*c - b^2)^9)^(1/2) - 615*a*b^13*c - 246*a^2*b^2*c^2*(-(4*a*c - b^2)^9
)^(1/2) + 165*a*b^4*c*(-(4*a*c - b^2)^9)^(1/2))/(32*(a^7*b^12*e^2*f^8 + 4096*a^13*c^6*e^2*f^8 + 240*a^9*b^8*c^
2*e^2*f^8 - 1280*a^10*b^6*c^3*e^2*f^8 + 3840*a^11*b^4*c^4*e^2*f^8 - 6144*a^12*b^2*c^5*e^2*f^8 - 24*a^8*b^10*c*
e^2*f^8)))^(1/2)*(x*(256*a^15*b^13*c^2*e^14*f^20 - 6144*a^16*b^11*c^3*e^14*f^20 + 61440*a^17*b^9*c^4*e^14*f^20
- 327680*a^18*b^7*c^5*e^14*f^20 + 983040*a^19*b^5*c^6*e^14*f^20 - 1572864*a^20*b^3*c^7*e^14*f^20 + 1048576*a^
21*b*c^8*e^14*f^20) + 1048576*a^21*b*c^8*d*e^13*f^20 + 256*a^15*b^13*c^2*d*e^13*f^20 - 6144*a^16*b^11*c^3*d*e^
13*f^20 + 61440*a^17*b^9*c^4*d*e^13*f^20 - 327680*a^18*b^7*c^5*d*e^13*f^20 + 983040*a^19*b^5*c^6*d*e^13*f^20 -
1572864*a^20*b^3*c^7*d*e^13*f^20) + 917504*a^19*c^9*e^12*f^16 - 320*a^12*b^14*c^2*e^12*f^16 + 7936*a^13*b^12*
c^3*e^12*f^16 - 82816*a^14*b^10*c^4*e^12*f^16 + 468480*a^15*b^8*c^5*e^12*f^16 - 1536000*a^16*b^6*c^6*e^12*f^16
+ 2867200*a^17*b^4*c^7*e^12*f^16 - 2719744*a^18*b^2*c^8*e^12*f^16) - x*(401408*a^16*c^10*e^12*f^12 - 400*a^9*
b^14*c^3*e^12*f^12 + 9440*a^10*b^12*c^4*e^12*f^...
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