3.7.38 \(\int \frac {(d f+e f x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx\) [638]

Optimal. Leaf size=202 \[ \frac {f^4 x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1156, 1136, 1180, 211} \begin {gather*} -\frac {f^4 \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 {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} e \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {f^4 \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 {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} e \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {f^4 x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^4*x)/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*f^4*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*f^4*ArcTan[
(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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 1136

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

Rubi steps

\begin {align*} \int \frac {(d f+e f x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {f^4 \text {Subst}\left (\int \frac {x^4}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {f^4 x}{c}-\frac {f^4 \text {Subst}\left (\int \frac {a+b x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{c e}\\ &=\frac {f^4 x}{c}-\frac {\left (\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4\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 c e}-\frac {\left (\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4\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 c e}\\ &=\frac {f^4 x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 222, normalized size = 1.10 \begin {gather*} \frac {f^4 \left (2 \sqrt {c} (d+e x)-\frac {\sqrt {2} \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 {\sqrt {2} \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}}}\right )}{2 c^{3/2} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^4*(2*Sqrt[c]*(d + e*x) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/S
qrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(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 + Sqr
t[b^2 - 4*a*c]])))/(2*c^(3/2)*e)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.20, size = 162, normalized size = 0.80

method	result	size
default	\(f^{4} \left (\frac {x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} b \,e^{2}-2 \textit {\_R} b d e -d^{2} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 c e}\right )\)	\(162\)
risch	\(\frac {f^{4} x}{c}+\frac {f^{4} \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} b \,e^{2}-2 \textit {\_R} b d e -d^{2} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 c e}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^4*(x/c+1/2/c/e*sum((-_R^2*b*e^2-2*_R*b*d*e-b*d^2-a)/(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)))

________________________________________________________________________________________

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

[Out]

f^4*x/c - f^4*integrate((b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a)/(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)/c

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (166) = 332\).
time = 0.39, size = 1282, normalized size = 6.35 \begin {gather*} \frac {2 \, f^{4} x - \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} f^{12} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d f^{12} + \sqrt {\frac {1}{2}} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} f^{8} e - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} f^{12} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d f^{12} - \sqrt {\frac {1}{2}} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} f^{8} e - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} f^{12} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d f^{12} + \sqrt {\frac {1}{2}} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} f^{8} e + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} f^{12} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d f^{12} - \sqrt {\frac {1}{2}} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} f^{8} e + \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} \sqrt {-\frac {{\left ({\left (b^{3} - 3 \, a b c\right )} f^{8} - \sqrt {\frac {{\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} f^{16}}{b^{2} c^{6} - 4 \, a c^{7}}} {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*f^4*x - sqrt(1/2)*c*sqrt(-((b^3 - 3*a*b*c)*f^8 + sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c
^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*f^12*x*e - 2*(a*b^2 - a^2*c)*d*f^
12 + sqrt(1/2)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*f^8*e - sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7
))*(b^3*c^3 - 4*a*b*c^4)*e)*sqrt(-((b^3 - 3*a*b*c)*f^8 + sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*
c^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))) + sqrt(1/2)*c*sqrt(-((b^3 - 3*a*b*c)*f^8 + sqrt((b^4 -
2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2
- a^2*c)*f^12*x*e - 2*(a*b^2 - a^2*c)*d*f^12 - sqrt(1/2)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*f^8*e - sqrt((b^4 - 2
*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^3*c^3 - 4*a*b*c^4)*e)*sqrt(-((b^3 - 3*a*b*c)*f^8 + sqrt((b^4
- 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))) - sqrt(1/2)
*c*sqrt(-((b^3 - 3*a*b*c)*f^8 - sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4)
)*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*f^12*x*e - 2*(a*b^2 - a^2*c)*d*f^12 + sqrt(1/2)*((b^4 - 5
*a*b^2*c + 4*a^2*c^2)*f^8*e + sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^3*c^3 - 4*a*b*c^4)
*e)*sqrt(-((b^3 - 3*a*b*c)*f^8 - sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4
))*e^(-2)/(b^2*c^3 - 4*a*c^4))) + sqrt(1/2)*c*sqrt(-((b^3 - 3*a*b*c)*f^8 - sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^
16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*f^12*x*e - 2*(
a*b^2 - a^2*c)*d*f^12 - sqrt(1/2)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*f^8*e + sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f^16
/(b^2*c^6 - 4*a*c^7))*(b^3*c^3 - 4*a*b*c^4)*e)*sqrt(-((b^3 - 3*a*b*c)*f^8 - sqrt((b^4 - 2*a*b^2*c + a^2*c^2)*f
^16/(b^2*c^6 - 4*a*c^7))*(b^2*c^3 - 4*a*c^4))*e^(-2)/(b^2*c^3 - 4*a*c^4))))/c

________________________________________________________________________________________

Sympy [A]
time = 2.09, size = 219, normalized size = 1.08 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{5} e^{4} - 128 a b^{2} c^{4} e^{4} + 16 b^{4} c^{3} e^{4}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} e^{2} f^{8} - 28 a b^{3} c e^{2} f^{8} + 4 b^{5} e^{2} f^{8}\right ) + a^{3} f^{16}, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a b c^{4} e^{3} - 8 t^{3} b^{3} c^{3} e^{3} - 4 t a^{2} c^{2} e f^{8} + 8 t a b^{2} c e f^{8} - 2 t b^{4} e f^{8} + a^{2} c d f^{12} - a b^{2} d f^{12}}{a^{2} c e f^{12} - a b^{2} e f^{12}} \right )} \right )\right )} + \frac {f^{4} x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(256*a**2*c**5*e**4 - 128*a*b**2*c**4*e**4 + 16*b**4*c**3*e**4) + _t**2*(48*a**2*b*c**2*e**2*f**
8 - 28*a*b**3*c*e**2*f**8 + 4*b**5*e**2*f**8) + a**3*f**16, Lambda(_t, _t*log(x + (32*_t**3*a*b*c**4*e**3 - 8*
_t**3*b**3*c**3*e**3 - 4*_t*a**2*c**2*e*f**8 + 8*_t*a*b**2*c*e*f**8 - 2*_t*b**4*e*f**8 + a**2*c*d*f**12 - a*b*
*2*d*f**12)/(a**2*c*e*f**12 - a*b**2*e*f**12)))) + f**4*x/c

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1245 vs. \(2 (166) = 332\).
time = 4.79, size = 1245, normalized size = 6.16 \begin {gather*} \frac {f^{4} x}{c} + \frac {{\left (\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 f^{4} e^{6} - 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 d f^{4} e^{5} + b d^{2} f^{4} e^{4} + a f^{4} e^{4}\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 f^{4} e^{6} - 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 d f^{4} e^{5} + b d^{2} f^{4} e^{4} + a f^{4} e^{4}\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 f^{4} e^{6} - 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 d f^{4} e^{5} + b d^{2} f^{4} e^{4} + a f^{4} e^{4}\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 f^{4} e^{6} - 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 d f^{4} e^{5} + b d^{2} f^{4} e^{4} + a f^{4} e^{4}\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 )}}\right )} e^{\left (-4\right )}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^4*x/c + 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*f^4*e^6 - 2*(d*e^(-
1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*f^4*e^5 + b*d^2*f^4*e^4 + a*f^4*e^4)*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*f^4*e^
6 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*f^4*e^5 + b*d^2*f^4*e^4 + a*f
^4*e^4)*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*f^4*e^6 - 2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*f^4*e^5 + b*d^2*
f^4*e^4 + a*f^4*e^4)*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*f^4*e^6 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*f^4
*e^5 + b*d^2*f^4*e^4 + a*f^4*e^4)*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))))*e^(-4)/c

________________________________________________________________________________________

Mupad [B]
time = 1.34, size = 2500, normalized size = 12.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((2*b^4*d*e^11*f^8 + 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*e^11*f^8)/c + ((16*a^2*c^3*e^12*f^4 - 4*a*b^2*c^
2*e^12*f^4)/c + ((8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5
*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))
/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) +
12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^
2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 - 4*a*b^2*c*e^12*f^8))/c)*(-(b^5*f^8 + b^2*f^8*(-
(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e
^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*1i + ((2*b^4*d*e^11*f^8 + 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*e^11*
f^8)/c - ((16*a^2*c^3*e^12*f^4 - 4*a*b^2*c^2*e^12*f^4)/c - ((8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4
*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^
3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^
5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2)
)/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 - 4*a*
b^2*c*e^12*f^8))/c)*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8
*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*1i)/(((2*b^4*d*e^11*f^8
+ 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*e^11*f^8)/c + ((16*a^2*c^3*e^12*f^4 - 4*a*b^2*c^2*e^12*f^4)/c + ((8*b^3*
c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5*f^8 + b^2*f^8*(-(4*a*c
- b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^
4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b
^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*
x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 - 4*a*b^2*c*e^12*f^8))/c)*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) +
12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b
^2*c^4*e^2)))^(1/2) - ((2*b^4*d*e^11*f^8 + 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*e^11*f^8)/c - ((16*a^2*c^3*e^12*
f^4 - 4*a*b^2*c^2*e^12*f^4)/c - ((8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*
e^14))/c)*(-(b^5*f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c
- b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^5*f^8 + b^2*f^8*(-(4*a*c -
b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*
c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 - 4*a*b^2*c*e^12*f^8))/c)*(-(b^5*
f^8 + b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/
(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*a^2*b*e^10*f^12)/c))*(-(b^5*f^8 + b^2*f^8*(-(
4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 - a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^
2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*2i + atan((((2*b^4*d*e^11*f^8 + 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*
e^11*f^8)/c + ((16*a^2*c^3*e^12*f^4 - 4*a*b^2*c^2*e^12*f^4)/c + ((8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2
*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5*f^8 - b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7
*a*b^3*c*f^8 + a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*
(-(b^5*f^8 - b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 + a*c*f^8*(-(4*a*c - b^2)^3)^
(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 -
4*a*b^2*c*e^12*f^8))/c)*(-(b^5*f^8 - b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 + a*
c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*1i + ((2*b^4*d*e^1
1*f^8 + 4*a^2*c^2*d*e^11*f^8 - 8*a*b^2*c*d*e^11*f^8)/c - ((16*a^2*c^3*e^12*f^4 - 4*a*b^2*c^2*e^12*f^4)/c - ((8
*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5*f^8 - b^2*f^8*(-(4
*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 + a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2
+ b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^5*f^8 - b^2*f^8*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*f^8 -
7*a*b^3*c*f^8 + a*c*f^8*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)
+ (2*x*(b^4*e^12*f^8 + 2*a^2*c^2*e^12*f^8 - 4*a*b^2*c*e^12*f^8))/c)*(-(b^5*f^8 - b^2*f^8*(-(4*a*c - b^2)^3)^(1
/2) + 12*a^2*b*c^2*f^8 - 7*a*b^3*c*f^8 + a*c*f^...

________________________________________________________________________________________