\(\int \frac {d+e x}{a+b x^2+c x^4} \, dx\) [20]

Optimal result

Integrand size = 20, antiderivative size = 189 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1687, 12, 1107, 211, 1121, 632, 212} \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]

[In]

Int[(d + e*x)/(a + b*x^2 + c*x^4),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1107

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

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\frac {\frac {2 \sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}+e \left (\log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )-\log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )\right )}{2 \sqrt {b^2-4 a c}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}\right )}{2}\)	\(43\)
default	\(4 c \left (\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {e \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {d \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 a c -2 b^{2}}-\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {e \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}-\frac {d \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 a c -2 b^{2}}\right )\)	\(200\)

[In]

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

[Out]

1/2*sum((_R*e+d)/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.16 (sec) , antiderivative size = 398481, normalized size of antiderivative = 2108.37 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\int { \frac {e x + d}{c x^{4} + b x^{2} + a} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1342 vs. \(2 (149) = 298\).

Time = 1.40 (sec) , antiderivative size = 1342, normalized size of antiderivative = 7.10 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(
2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sq
rt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^2
+ 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c
+ 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*d*arctan(2*sq
rt(1/2)*x/sqrt((b + sqrt(b^2 - 4*a*c))/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^
2*c^2 - 4*a^2*c^3)*abs(c)) + 1/4*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*b^2*c^2 - 16*a*b^2*c^2 + 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 - 8*a
*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c - sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*a*c^2 - 2*(b^2
- 4*a*c)*b*c^2)*d*arctan(2*sqrt(1/2)*x/sqrt((b - sqrt(b^2 - 4*a*c))/c))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16
*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2*c^3)*abs(c)) - 1/2*(b^2*c^2 - 4*a*c^3 - 2*b*c^3 + c^4)*sqrt(b^2 - 4
*a*c)*e*log(x^2 + 1/2*(b - sqrt(b^2 - 4*a*c))/c)/((b^4 - 8*a*b^2*c - 2*b^3*c + 16*a^2*c^2 + 8*a*b*c^2 + b^2*c^
2 - 4*a*c^3)*c^2) + 1/4*(b^5*c - 8*a*b^3*c^2 - 2*b^4*c^2 + 16*a^2*b*c^3 + 8*a*b^2*c^3 + b^3*c^3 - 4*a*b*c^4 +
(b^4*c - 6*a*b^2*c^2 - 2*b^3*c^2 + 8*a^2*c^3 + 4*a*b*c^3 + b^2*c^3 - 2*a*c^4)*sqrt(b^2 - 4*a*c))*e*log(x^2 + 1
/2*(b + sqrt(b^2 - 4*a*c))/c)/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2
*c^3)*c^2)

Mupad [B] (verification not implemented)

Time = 8.36 (sec) , antiderivative size = 1308, normalized size of antiderivative = 6.92 \[ \int \frac {d+e x}{a+b x^2+c x^4} \, dx=\sum _{k=1}^4\ln \left (c^2\,\left (d\,e^2+e^3\,x+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,d\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,b^3\,x\,8-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,d\,16+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,e^2\,x\,2-\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,c\,d^2\,x\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,e\,x\,4+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,d\,e\,4+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,a\,b\,c\,x\,32+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,e\,x\,16\right )\right )\,\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right ) \]

[In]

int((d + e*x)/(a + b*x^2 + c*x^4),x)

[Out]

symsum(log(c^2*(d*e^2 + e^3*x + 4*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 -
32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e
^4, z, k)^2*b^2*d - 8*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^
2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^3*
b^3*x - 16*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a
*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^2*a*c*d + 2*r
oot(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2
- 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)*b*e^2*x - 4*root(128*a^2*
b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2
*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)*c*d^2*x - 4*root(128*a^2*b^2*c*z^4 -
256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*
c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^2*b^2*e*x + 4*root(128*a^2*b^2*c*z^4 - 256*a^3*c^
2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z
- 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)*b*d*e + 32*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*
b^4*z^4 + 16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e
*z - b*d^2*e^2 - c*d^4 - a*e^4, z, k)^3*a*b*c*x + 16*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 +
16*a*b*c*d^2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^
2*e^2 - c*d^4 - a*e^4, z, k)^2*a*c*e*x))*root(128*a^2*b^2*c*z^4 - 256*a^3*c^2*z^4 - 16*a*b^4*z^4 + 16*a*b*c*d^
2*z^2 - 32*a^2*c*e^2*z^2 + 8*a*b^2*e^2*z^2 - 4*b^3*d^2*z^2 + 16*a*c*d^2*e*z - 4*b^2*d^2*e*z - b*d^2*e^2 - c*d^
4 - a*e^4, z, k), k, 1, 4)