\(\int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx\) [165]

Optimal result

Integrand size = 24, antiderivative size = 136 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]

[Out]

1/12*c*arctan(-1+6^(1/4)*x)*6^(1/4)+1/12*c*arctan(1+6^(1/4)*x)*6^(1/4)+1/12*d*ln(3*x^4+2)+1/24*c*ln(-6^(3/4)*x
+3*x^2+6^(1/2))*6^(1/4)-1/24*c*ln(6^(3/4)*x+3*x^2+6^(1/2))*6^(1/4)+1/12*b*arctan(1/2*x^2*6^(1/2))*6^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1608, 1845, 303, 1176, 631, 210, 1179, 642, 1262, 649, 209, 266} \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac {c \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}-\frac {c \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (3 x^4+2\right ) \]

[In]

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

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (c*ArcTan[1 - 6^(1/4)*x])/(2*6^(3/4)) + (c*ArcTan[1 + 6^(1/4)*x])/(2*6
^(3/4)) + (c*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(4*6^(3/4)) - (c*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(4*6^(3/4))
+ (d*Log[2 + 3*x^4])/12

Rule 209

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

Rule 210

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 631

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1176

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

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1845

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(c*x)^(m + ii)*((Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (b+c x+d x^2\right )}{2+3 x^4} \, dx \\ & = \int \left (\frac {c x^2}{2+3 x^4}+\frac {x \left (b+d x^2\right )}{2+3 x^4}\right ) \, dx \\ & = c \int \frac {x^2}{2+3 x^4} \, dx+\int \frac {x \left (b+d x^2\right )}{2+3 x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {b+d x}{2+3 x^2} \, dx,x,x^2\right )-\frac {c \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {3}}+\frac {c \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {3}} \\ & = \frac {1}{2} b \text {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right )+\frac {1}{12} c \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{12} c \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {c \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{4\ 6^{3/4}}+\frac {c \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{4\ 6^{3/4}}+\frac {1}{2} d \text {Subst}\left (\int \frac {x}{2+3 x^2} \, dx,x,x^2\right ) \\ & = \frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right )+\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}-\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}} \\ & = \frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \left (-2 \sqrt [4]{6} \left (\sqrt [4]{6} b+c\right ) \arctan \left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (-\sqrt [4]{6} b+c\right ) \arctan \left (1+\sqrt [4]{6} x\right )+\sqrt [4]{6} c \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\sqrt [4]{6} c \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+2 d \log \left (2+3 x^4\right )\right ) \]

[In]

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

[Out]

(-2*6^(1/4)*(6^(1/4)*b + c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(-(6^(1/4)*b) + c)*ArcTan[1 + 6^(1/4)*x] + 6^(1/
4)*c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/2
4

Maple [C] (verified)

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

Time = 1.48 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.28

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +\textit {\_R}^{2} c +\textit {\_R} b \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\)	\(38\)
default	\(\frac {b \arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{12}+\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\)	\(121\)
meijerg	\(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}+\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{2}\right )}{12}\)	\(201\)

[In]

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

[Out]

1/12*sum((_R^3*d+_R^2*c+_R*b)/_R^3*ln(x-_R),_R=RootOf(3*_Z^4+2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 18086, normalized size of antiderivative = 132.99 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [A] (verification not implemented)

Time = 0.83 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.39 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (82944 t^{4} - 27648 t^{3} d + t^{2} \cdot \left (1728 b^{2} + 3456 d^{2}\right ) + t \left (- 288 b^{2} d + 288 b c^{2} - 192 d^{3}\right ) + 9 b^{4} + 12 b^{2} d^{2} - 24 b c^{2} d + 6 c^{4} + 4 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 3456 t^{3} c^{2} + 864 t^{2} b^{3} + 864 t^{2} c^{2} d - 144 t b^{3} d - 108 t b^{2} c^{2} - 72 t c^{2} d^{2} + 9 b^{5} + 6 b^{3} d^{2} + 9 b^{2} c^{2} d - 9 b c^{4} + 2 c^{2} d^{3}}{18 b^{4} c - 3 c^{5}} \right )} \right )\right )} \]

[In]

integrate((d*x**3+c*x**2+b*x)/(3*x**4+2),x)

[Out]

RootSum(82944*_t**4 - 27648*_t**3*d + _t**2*(1728*b**2 + 3456*d**2) + _t*(-288*b**2*d + 288*b*c**2 - 192*d**3)
+ 9*b**4 + 12*b**2*d**2 - 24*b*c**2*d + 6*c**4 + 4*d**4, Lambda(_t, _t*log(x + (-3456*_t**3*c**2 + 864*_t**2*
b**3 + 864*_t**2*c**2*d - 144*_t*b**3*d - 108*_t*b**2*c**2 - 72*_t*c**2*d**2 + 9*b**5 + 6*b**3*d**2 + 9*b**2*c
**2*d - 9*b*c**4 + 2*c**2*d**3)/(18*b**4*c - 3*c**5))))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.28 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{72} \, \sqrt {3} \sqrt {2} {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} c - 6 \, b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \, \sqrt {3} \sqrt {2} {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} c + 6 \, b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d - \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d + \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]

[In]

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

[Out]

1/72*sqrt(3)*sqrt(2)*(3^(3/4)*2^(3/4)*c - 6*b)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1
/72*sqrt(3)*sqrt(2)*(3^(3/4)*2^(3/4)*c + 6*b)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/
72*3^(3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d - sqrt(3)*c)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/72*3^(3/
4)*2^(1/4)*(3^(1/4)*2^(3/4)*d + sqrt(3)*c)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=-\frac {1}{12} \, {\left (\sqrt {6} b - 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \, {\left (\sqrt {6} b + 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \, {\left (6^{\frac {1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{24} \, {\left (6^{\frac {1}{4}} c + 2 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]

[In]

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

[Out]

-1/12*(sqrt(6)*b - 6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/12*(sqrt(6)*b +
6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) - 1/24*(6^(1/4)*c - 2*d)*log(x^2 + sqrt
(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/24*(6^(1/4)*c + 2*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))

Mupad [B] (verification not implemented)

Time = 10.47 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.21 \[ \int \frac {b x+c x^2+d x^3}{2+3 x^4} \, dx=\sum _{k=1}^4\ln \left (-\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right )\,\left (144\,b\,c+x\,\left (144\,b\,d-72\,c^2\right )-\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right )\,b\,x\,864\right )+x\,\left (9\,b^3+6\,b\,d^2-6\,c^2\,d\right )-6\,c^3+12\,b\,c\,d\right )\,\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (1728\,b^2+3456\,d^2\right )}{82944}-\frac {z\,\left (-288\,b\,c^2+288\,b^2\,d+192\,d^3\right )}{82944}-\frac {b\,c^2\,d}{3456}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {c^4}{13824}+\frac {b^4}{9216},z,k\right ) \]

[In]

int((b*x + c*x^2 + d*x^3)/(3*x^4 + 2),x)

[Out]

symsum(log(x*(6*b*d^2 - 6*c^2*d + 9*b^3) - root(z^4 - (d*z^3)/3 + (z^2*(1728*b^2 + 3456*d^2))/82944 - (z*(- 28
8*b*c^2 + 288*b^2*d + 192*d^3))/82944 - (b*c^2*d)/3456 + (b^2*d^2)/6912 + d^4/20736 + c^4/13824 + b^4/9216, z,
k)*(144*b*c + x*(144*b*d - 72*c^2) - 864*root(z^4 - (d*z^3)/3 + (z^2*(1728*b^2 + 3456*d^2))/82944 - (z*(- 288
*b*c^2 + 288*b^2*d + 192*d^3))/82944 - (b*c^2*d)/3456 + (b^2*d^2)/6912 + d^4/20736 + c^4/13824 + b^4/9216, z,
k)*b*x) - 6*c^3 + 12*b*c*d)*root(z^4 - (d*z^3)/3 + (z^2*(1728*b^2 + 3456*d^2))/82944 - (z*(- 288*b*c^2 + 288*b
^2*d + 192*d^3))/82944 - (b*c^2*d)/3456 + (b^2*d^2)/6912 + d^4/20736 + c^4/13824 + b^4/9216, z, k), k, 1, 4)