3.7.45 \(\int \frac {1}{(d f+e f x)^4 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [645]
Optimal. Leaf size=236 \[ -\frac {1}{3 a e f^4 (d+e x)^3}+\frac {b}{a^2 e f^4 (d+e x)}+\frac {\sqrt {c} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}} e f^4}+\frac {\sqrt {c} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+\sqrt {b^2-4 a c}} e f^4} \]
[Out]
-1/3/a/e/f^4/(e*x+d)^3+b/a^2/e/f^4/(e*x+d)+1/2*arctan((e*x+d)*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^
(1/2)*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^2/e/f^4*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/2*arctan((e*x+d)*2^
(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b+(2*a*c-b^2)/(-4*a*c+b^2)^(1/2))/a^2/e/f^4*2^(1/2)/(b+(-
4*a*c+b^2)^(1/2))^(1/2)
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Rubi [A]
time = 0.34, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1156, 1137,
1295, 1180, 211} \begin {gather*} \frac {\sqrt {c} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 e f^4 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b-\frac {b^2-2 a c}{\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 )}{\sqrt {2} a^2 e f^4 \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {b}{a^2 e f^4 (d+e x)}-\frac {1}{3 a e f^4 (d+e x)^3} \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)),x]
[Out]
-1/3*1/(a*e*f^4*(d + e*x)^3) + b/(a^2*e*f^4*(d + e*x)) + (Sqrt[c]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan
[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^4) + (
Sqrt[c]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])
/(Sqrt[2]*a^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 1137
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2 +
c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[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 )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e f^4}\\ &=-\frac {1}{3 a e f^4 (d+e x)^3}+\frac {\text {Subst}\left (\int \frac {-3 b-3 c x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{3 a e f^4}\\ &=-\frac {1}{3 a e f^4 (d+e x)^3}+\frac {b}{a^2 e f^4 (d+e x)}-\frac {\text {Subst}\left (\int \frac {-3 \left (b^2-a c\right )-3 b c x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{3 a^2 e f^4}\\ &=-\frac {1}{3 a e f^4 (d+e x)^3}+\frac {b}{a^2 e f^4 (d+e x)}+\frac {\left (c \left (b-\frac {b^2-2 a c}{\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 )}{2 a^2 e f^4}+\frac {\left (c \left (b+\frac {b^2-2 a c}{\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 )}{2 a^2 e f^4}\\ &=-\frac {1}{3 a e f^4 (d+e x)^3}+\frac {b}{a^2 e f^4 (d+e x)}+\frac {\sqrt {c} \left (b+\frac {b^2-2 a 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 )}{\sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}} e f^4}+\frac {\sqrt {c} \left (b-\frac {b^2-2 a 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 )}{\sqrt {2} a^2 \sqrt {b+\sqrt {b^2-4 a c}} e f^4}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 238, normalized size = 1.01 \begin {gather*} \frac {-\frac {2 a}{(d+e x)^3}+\frac {6 b}{d+e x}+\frac {3 \sqrt {2} \sqrt {c} \left (b^2-2 a c+b \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 )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-b^2+2 a c+b \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 )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{6 a^2 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)),x]
[Out]
((-2*a)/(d + e*x)^3 + (6*b)/(d + e*x) + (3*Sqrt[2]*Sqrt[c]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]
*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]
*Sqrt[c]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])
/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(6*a^2*e*f^4)
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.20, size = 192, normalized size = 0.81
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| method |
result |
size |
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| default |
\(\frac {-\frac {1}{3 a e \left (e x +d \right )^{3}}+\frac {b}{a^{2} e \left (e x +d \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} b c \,e^{2}+2 \textit {\_R} b c d e +b c \,d^{2}-a c +b^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a^{2} e}}{f^{4}}\) |
\(192\) |
| risch |
\(\frac {\frac {b e \,x^{2}}{a^{2}}+\frac {2 b d x}{a^{2}}-\frac {-3 d^{2} b +a}{3 e \,a^{2}}}{f^{4} \left (e x +d \right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 f^{16} e^{4} c^{2} a^{7}-8 a^{6} b^{2} c \,e^{4} f^{16}+a^{5} b^{4} e^{4} f^{16}\right ) \textit {\_Z}^{4}+\left (-20 b \,f^{8} e^{2} c^{3} a^{3}+25 b^{3} f^{8} e^{2} c^{2} a^{2}-9 b^{5} f^{8} e^{2} c a +b^{7} f^{8} e^{2}\right ) \textit {\_Z}^{2}+c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{7} c^{2} e^{5} f^{16}-22 a^{6} b^{2} c \,e^{5} f^{16}+3 a^{5} b^{4} e^{5} f^{16}\right ) \textit {\_R}^{4}+\left (-43 a^{3} b \,c^{3} e^{3} f^{8}+51 a^{2} b^{3} c^{2} e^{3} f^{8}-18 a \,b^{5} c \,e^{3} f^{8}+2 b^{7} e^{3} f^{8}\right ) \textit {\_R}^{2}+2 e \,c^{5}\right ) x +\left (40 a^{7} c^{2} d \,e^{4} f^{16}-22 a^{6} b^{2} c d \,e^{4} f^{16}+3 a^{5} b^{4} d \,e^{4} f^{16}\right ) \textit {\_R}^{4}+\left (-8 a^{5} b \,c^{2} e^{3} f^{12}+6 a^{4} b^{3} c \,e^{3} f^{12}-a^{3} b^{5} e^{3} f^{12}\right ) \textit {\_R}^{3}+\left (-43 a^{3} b \,c^{3} d \,e^{2} f^{8}+51 a^{2} b^{3} c^{2} d \,e^{2} f^{8}-18 a \,b^{5} c d \,e^{2} f^{8}+2 b^{7} d \,e^{2} f^{8}\right ) \textit {\_R}^{2}+a^{2} c^{4} e \,f^{4} \textit {\_R} +2 c^{5} d \right )\right )}{2}\) |
\(469\) |
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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),x,method=_RETURNVERBOSE)
[Out]
1/f^4*(-1/3/a/e/(e*x+d)^3+b/a^2/e/(e*x+d)+1/2/a^2/e*sum((_R^2*b*c*e^2+2*_R*b*c*d*e+b*c*d^2-a*c+b^2)/(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),x, algorithm="maxima")
[Out]
1/3*(3*b*x^2*e^2 + 6*b*d*x*e + 3*b*d^2 - a)/(a^2*f^4*x^3*e^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^2*d^2*f^4*x*e^2 + a^2
*d^3*f^4*e) + integrate((b*c*x^2*e^2 + 2*b*c*d*x*e + b*c*d^2 + b^2 - a*c)/(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^2*f^4)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2138 vs.
\(2 (200) = 400\).
time = 0.41, size = 2138, normalized size = 9.06 \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),x, algorithm="fricas")
[Out]
1/6*(6*b*x^2*e^2 + 12*b*d*x*e - 3*sqrt(1/2)*(a^2*f^4*x^3*e^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^2*d^2*f^4*x*e^2 + a^2
*d^3*f^4*e)*sqrt(((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((
a^10*b^2 - 4*a^11*c)*f^16)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^2 - 4*a^6*c)*f^8))*e^(-1)*log(2*(b^4*c^3
- 3*a*b^2*c^4 + a^2*c^5)*x*e + sqrt(1/2)*((a^5*b^5 - 7*a^6*b^3*c + 12*a^7*b*c^2)*f^12*sqrt((b^8 - 6*a*b^6*c +
11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16))*e + (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2
- 17*a^3*b^2*c^3 + 4*a^4*c^4)*f^4*e)*sqrt(((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*
a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^2 - 4*a^6*c)*f^8
))*e^(-1) + 2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*d) + 3*sqrt(1/2)*(a^2*f^4*x^3*e^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^
2*d^2*f^4*x*e^2 + a^2*d^3*f^4*e)*sqrt(((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*
b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a^5*b^2 - 4*a^6*c)*f^8))*e
^(-1)*log(2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*x*e - sqrt(1/2)*((a^5*b^5 - 7*a^6*b^3*c + 12*a^7*b*c^2)*f^12*sqr
t((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16))*e + (b^8 - 8*a*b^
6*c + 20*a^2*b^4*c^2 - 17*a^3*b^2*c^3 + 4*a^4*c^4)*f^4*e)*sqrt(((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c
+ 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) - b^5 + 5*a*b^3*c - 5*a^2*b*c^2)/((a
^5*b^2 - 4*a^6*c)*f^8))*e^(-1) + 2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*d) + 6*b*d^2 + 3*sqrt(1/2)*(a^2*f^4*x^3*e
^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^2*d^2*f^4*x*e^2 + a^2*d^3*f^4*e)*sqrt(-((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a
*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) + b^5 - 5*a*b^3*c + 5*a^2*b*c
^2)*e^(-2)/((a^5*b^2 - 4*a^6*c)*f^8))*log(2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*x*e + 2*(b^4*c^3 - 3*a*b^2*c^4 +
a^2*c^5)*d + sqrt(1/2)*((a^5*b^5 - 7*a^6*b^3*c + 12*a^7*b*c^2)*f^12*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 -
6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16))*e - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 17*a^3*b^2*c^3
+ 4*a^4*c^4)*f^4*e)*sqrt(-((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^
4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^(-2)/((a^5*b^2 - 4*a^6*c)*f^8))) - 3*s
qrt(1/2)*(a^2*f^4*x^3*e^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^2*d^2*f^4*x*e^2 + a^2*d^3*f^4*e)*sqrt(-((a^5*b^2 - 4*a^6
*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) + b^5
- 5*a*b^3*c + 5*a^2*b*c^2)*e^(-2)/((a^5*b^2 - 4*a^6*c)*f^8))*log(2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*x*e + 2*(
b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*d - sqrt(1/2)*((a^5*b^5 - 7*a^6*b^3*c + 12*a^7*b*c^2)*f^12*sqrt((b^8 - 6*a*b^
6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16))*e - (b^8 - 8*a*b^6*c + 20*a^2*b^
4*c^2 - 17*a^3*b^2*c^3 + 4*a^4*c^4)*f^4*e)*sqrt(-((a^5*b^2 - 4*a^6*c)*f^8*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c
^2 - 6*a^3*b^2*c^3 + a^4*c^4)/((a^10*b^2 - 4*a^11*c)*f^16)) + b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^(-2)/((a^5*b^2
- 4*a^6*c)*f^8))) - 2*a)/(a^2*f^4*x^3*e^4 + 3*a^2*d*f^4*x^2*e^3 + 3*a^2*d^2*f^4*x*e^2 + a^2*d^3*f^4*e)
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Sympy [A]
time = 103.09, size = 411, normalized size = 1.74 \begin {gather*} \frac {- a + 3 b d^{2} + 6 b d e x + 3 b e^{2} x^{2}}{3 a^{2} d^{3} e f^{4} + 9 a^{2} d^{2} e^{2} f^{4} x + 9 a^{2} d e^{3} f^{4} x^{2} + 3 a^{2} e^{4} f^{4} x^{3}} + \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{7} c^{2} e^{4} f^{16} - 128 a^{6} b^{2} c e^{4} f^{16} + 16 a^{5} b^{4} e^{4} f^{16}\right ) + t^{2} \left (- 80 a^{3} b c^{3} e^{2} f^{8} + 100 a^{2} b^{3} c^{2} e^{2} f^{8} - 36 a b^{5} c e^{2} f^{8} + 4 b^{7} e^{2} f^{8}\right ) + c^{5}, \left ( t \mapsto t \log {\left (x + \frac {- 96 t^{3} a^{7} b c^{2} e^{3} f^{12} + 56 t^{3} a^{6} b^{3} c e^{3} f^{12} - 8 t^{3} a^{5} b^{5} e^{3} f^{12} - 4 t a^{4} c^{4} e f^{4} + 32 t a^{3} b^{2} c^{3} e f^{4} - 40 t a^{2} b^{4} c^{2} e f^{4} + 16 t a b^{6} c e f^{4} - 2 t b^{8} e f^{4} + a^{2} c^{5} d - 3 a b^{2} c^{4} d + b^{4} c^{3} d}{a^{2} c^{5} e - 3 a b^{2} c^{4} e + b^{4} c^{3} e} \right )} \right )\right )} \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),x)
[Out]
(-a + 3*b*d**2 + 6*b*d*e*x + 3*b*e**2*x**2)/(3*a**2*d**3*e*f**4 + 9*a**2*d**2*e**2*f**4*x + 9*a**2*d*e**3*f**4
*x**2 + 3*a**2*e**4*f**4*x**3) + RootSum(_t**4*(256*a**7*c**2*e**4*f**16 - 128*a**6*b**2*c*e**4*f**16 + 16*a**
5*b**4*e**4*f**16) + _t**2*(-80*a**3*b*c**3*e**2*f**8 + 100*a**2*b**3*c**2*e**2*f**8 - 36*a*b**5*c*e**2*f**8 +
4*b**7*e**2*f**8) + c**5, Lambda(_t, _t*log(x + (-96*_t**3*a**7*b*c**2*e**3*f**12 + 56*_t**3*a**6*b**3*c*e**3
*f**12 - 8*_t**3*a**5*b**5*e**3*f**12 - 4*_t*a**4*c**4*e*f**4 + 32*_t*a**3*b**2*c**3*e*f**4 - 40*_t*a**2*b**4*
c**2*e*f**4 + 16*_t*a*b**6*c*e*f**4 - 2*_t*b**8*e*f**4 + a**2*c**5*d - 3*a*b**2*c**4*d + b**4*c**3*d)/(a**2*c*
*5*e - 3*a*b**2*c**4*e + b**4*c**3*e))))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1249 vs.
\(2 (200) = 400\).
time = 4.33, size = 1249, normalized size = 5.29 \begin {gather*} -\frac {\frac {{\left ({\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b c e^{2} - 2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b c d e + b c d^{2} + b^{2} - a c\right )} \log \left (d e^{\left (-1\right )} + x + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b c e^{2} - 2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b c d e + b c d^{2} + b^{2} - a c\right )} \log \left (d e^{\left (-1\right )} + x - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b c e^{2} - 2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b c d e + b c d^{2} + b^{2} - a c\right )} \log \left (d e^{\left (-1\right )} + x + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b c e^{2} - 2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b c d e + b c d^{2} + b^{2} - a c\right )} \log \left (d e^{\left (-1\right )} + x - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}}}{2 \, a^{2} f^{4}} + \frac {{\left (3 \, b x^{2} e^{2} + 6 \, b d x e + 3 \, b d^{2} - a\right )} e^{\left (-1\right )}}{3 \, {\left (x e + d\right )}^{3} a^{2} f^{4}} \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),x, algorithm="giac")
[Out]
-1/2*(((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*e^2 - 2*(d*e^(-1) + sqrt(1
/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*e + b*c*d^2 + b^2 - a*c)*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))) + ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*e^2 - 2*(d*e^(-1) - sqrt(
1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*e + b*c*d^2 + b^2 - a*c)*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))) + ((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*e^2 - 2*(d*e^(-1) + sqrt
(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*e + b*c*d^2 + b^2 - a*c)*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))) + ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*c*e^2 - 2*(d*e^(-1) - sqr
t(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*c*d*e + b*c*d^2 + b^2 - a*c)*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^2*f^4) + 1/3*(3*b*x^2*e^2 + 6*b*d*x*e + 3*b*d^2 - a)*e^(-1)/((x*e + d)^3*a^2*f^4)
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Mupad [B]
time = 3.17, size = 2500, normalized size = 10.59 \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)),x)
[Out]
((2*b*d*x)/a^2 - (a - 3*b*d^2)/(3*a^2*e) + (b*e*x^2)/a^2)/(d^3*f^4 + e^3*f^4*x^3 + 3*d*e^2*f^4*x^2 + 3*d^2*e*f
^4*x) - atan((((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3
)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2
*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c -
b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*
a^6*b^2*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(
4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f
^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(x*(8*a^10*b^3*c^2*e^14*f^20 - 32*a^11*b*c^3*e^14*f^20) - 32*a^11*b*c^3*d*e^
13*f^20 + 8*a^10*b^3*c^2*d*e^13*f^20) - 16*a^10*c^4*e^12*f^16 - 4*a^8*b^4*c^2*e^12*f^16 + 20*a^9*b^2*c^3*e^12*
f^16) + x*(4*a^8*c^5*e^12*f^12 + 2*a^6*b^4*c^3*e^12*f^12 - 8*a^7*b^2*c^4*e^12*f^12) + 4*a^8*c^5*d*e^11*f^12 +
2*a^6*b^4*c^3*d*e^11*f^12 - 8*a^7*b^2*c^4*d*e^11*f^12)*1i + ((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^
3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^
5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a
^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))
/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7
+ 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)
^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(x*(8*a^10*b^3*c^2*e^14*f^20 -
32*a^11*b*c^3*e^14*f^20) - 32*a^11*b*c^3*d*e^13*f^20 + 8*a^10*b^3*c^2*d*e^13*f^20) + 16*a^10*c^4*e^12*f^16 +
4*a^8*b^4*c^2*e^12*f^16 - 20*a^9*b^2*c^3*e^12*f^16) + x*(4*a^8*c^5*e^12*f^12 + 2*a^6*b^4*c^3*e^12*f^12 - 8*a^7
*b^2*c^4*e^12*f^12) + 4*a^8*c^5*d*e^11*f^12 + 2*a^6*b^4*c^3*d*e^11*f^12 - 8*a^7*b^2*c^4*d*e^11*f^12)*1i)/(((b^
4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*
c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2
)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9
*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)
))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1
/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e
^2*f^8)))^(1/2)*(x*(8*a^10*b^3*c^2*e^14*f^20 - 32*a^11*b*c^3*e^14*f^20) - 32*a^11*b*c^3*d*e^13*f^20 + 8*a^10*b
^3*c^2*d*e^13*f^20) - 16*a^10*c^4*e^12*f^16 - 4*a^8*b^4*c^2*e^12*f^16 + 20*a^9*b^2*c^3*e^12*f^16) + x*(4*a^8*c
^5*e^12*f^12 + 2*a^6*b^4*c^3*e^12*f^12 - 8*a^7*b^2*c^4*e^12*f^12) + 4*a^8*c^5*d*e^11*f^12 + 2*a^6*b^4*c^3*d*e^
11*f^12 - 8*a^7*b^2*c^4*d*e^11*f^12) - ((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 +
a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^
7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3
*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8
+ 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*
a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e
^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)*(x*(8*a^10*b^3*c^2*e^14*f^20 - 32*a^11*b*c^3*e^14*f
^20) - 32*a^11*b*c^3*d*e^13*f^20 + 8*a^10*b^3*c^2*d*e^13*f^20) + 16*a^10*c^4*e^12*f^16 + 4*a^8*b^4*c^2*e^12*f^
16 - 20*a^9*b^2*c^3*e^12*f^16) + x*(4*a^8*c^5*e^12*f^12 + 2*a^6*b^4*c^3*e^12*f^12 - 8*a^7*b^2*c^4*e^12*f^12) +
4*a^8*c^5*d*e^11*f^12 + 2*a^6*b^4*c^3*d*e^11*f^12 - 8*a^7*b^2*c^4*d*e^11*f^12) + 2*a^6*b*c^5*e^10*f^8))*((b^4
*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 20*a^3*b*c^3 - 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 9*a*b^5*c
- 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*c*e^2*f^8)))^(1/2)
*2i - atan(((-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3 + 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c - b^2)^3)
^(1/2) - 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*a^6*b^2*
c*e^2*f^8)))^(1/2)*((-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3 + 25*a^2*b^3*c^2 + a^2*c^2*(-(4*a*c -
b^2)^3)^(1/2) - 9*a*b^5*c - 3*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^2*f^8 + 16*a^7*c^2*e^2*f^8 - 8*
a^6*b^2*c*e^2*f^8)))^(1/2)*((x*(8*a^10*b^3*c^2*...
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