Integrand size = 20, antiderivative size = 126 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \text {arctanh}\left (\frac {2 \sqrt [4]{6} x}{2+\sqrt {6} x^2}\right )}{4\ 6^{3/4}} \] Output:
1/12*b*arctan(1/2*6^(1/2)*x^2)*6^(1/2)+1/24*(6^(1/2)*a+2*c)*arctan(-1+6^(1 /4)*x)*6^(1/4)+1/24*(6^(1/2)*a+2*c)*arctan(1+6^(1/4)*x)*6^(1/4)+1/24*(6^(1 /2)*a-2*c)*arctanh(2*6^(1/4)*x/(2+6^(1/2)*x^2))*6^(1/4)
Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\frac {-2 \left (\sqrt {6} a+2 \left (\sqrt [4]{6} b+c\right )\right ) \arctan \left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt {6} a-2 \sqrt [4]{6} b+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )-\left (\sqrt {6} a-2 c\right ) \left (\log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right )}{8\ 6^{3/4}} \] Input:
Integrate[(a + b*x + c*x^2)/(2 + 3*x^4),x]
Output:
(-2*(Sqrt[6]*a + 2*(6^(1/4)*b + c))*ArcTan[1 - 6^(1/4)*x] + 2*(Sqrt[6]*a - 2*6^(1/4)*b + 2*c)*ArcTan[1 + 6^(1/4)*x] - (Sqrt[6]*a - 2*c)*(Log[2 - 2*6 ^(1/4)*x + Sqrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(3/4))
Time = 0.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2415, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b x+c x^2}{3 x^4+2} \, dx\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \int \left (\frac {a+c x^2}{3 x^4+2}+\frac {b x}{3 x^4+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}\) |
Input:
Int[(a + b*x + c*x^2)/(2 + 3*x^4),x]
Output:
(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - ((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1 /4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x])/(4*6^(3/4) ) - ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + ((S qrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3/4))
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{2} c +\textit {\_R} b +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(34\) |
| default | \(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}+\frac {b \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right ) \sqrt {6}}{12}+\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}\) | \(203\) |
| meijerg | \(\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}+\frac {\sqrt {6}\, b \arctan \left (\frac {x^{2} \sqrt {3}\, \sqrt {2}}{2}\right )}{12}+\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\) | \(352\) |
Input:
int((c*x^2+b*x+a)/(3*x^4+2),x,method=_RETURNVERBOSE)
Output:
1/12*sum((_R^2*c+_R*b+a)/_R^3*ln(x-_R),_R=RootOf(3*_Z^4+2))
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 46609, normalized size of antiderivative = 369.91 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)/(3*x^4+2),x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (110) = 220\).
Time = 2.57 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.32 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (55296 t^{4} + t^{2} \cdot \left (2304 a c + 1152 b^{2}\right ) + t \left (- 288 a^{2} b + 192 b c^{2}\right ) + 9 a^{4} + 12 a^{2} c^{2} - 24 a b^{2} c + 6 b^{4} + 4 c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 13824 t^{3} a^{2} c + 27648 t^{3} a b^{2} + 9216 t^{3} c^{3} + 1728 t^{2} a^{3} b + 3456 t^{2} a b c^{2} - 2304 t^{2} b^{3} c + 216 t a^{5} - 576 t a^{3} c^{2} + 1296 t a^{2} b^{2} c + 288 t a b^{4} + 288 t a c^{4} + 288 t b^{2} c^{3} + 90 a^{4} b c - 90 a^{3} b^{3} + 60 a b^{3} c^{2} - 24 b^{5} c + 24 b c^{5}}{27 a^{6} - 18 a^{4} c^{2} + 144 a^{3} b^{2} c - 72 a^{2} b^{4} - 12 a^{2} c^{4} + 96 a b^{2} c^{3} - 48 b^{4} c^{2} + 8 c^{6}} \right )} \right )\right )} \] Input:
integrate((c*x**2+b*x+a)/(3*x**4+2),x)
Output:
RootSum(55296*_t**4 + _t**2*(2304*a*c + 1152*b**2) + _t*(-288*a**2*b + 192 *b*c**2) + 9*a**4 + 12*a**2*c**2 - 24*a*b**2*c + 6*b**4 + 4*c**4, Lambda(_ t, _t*log(x + (-13824*_t**3*a**2*c + 27648*_t**3*a*b**2 + 9216*_t**3*c**3 + 1728*_t**2*a**3*b + 3456*_t**2*a*b*c**2 - 2304*_t**2*b**3*c + 216*_t*a** 5 - 576*_t*a**3*c**2 + 1296*_t*a**2*b**2*c + 288*_t*a*b**4 + 288*_t*a*c**4 + 288*_t*b**2*c**3 + 90*a**4*b*c - 90*a**3*b**3 + 60*a*b**3*c**2 - 24*b** 5*c + 24*b*c**5)/(27*a**6 - 18*a**4*c**2 + 144*a**3*b**2*c - 72*a**2*b**4 - 12*a**2*c**4 + 96*a*b**2*c**3 - 48*b**4*c**2 + 8*c**6))))
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (92) = 184\).
Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.48 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{24} \, {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} a - 2 \, \sqrt {3} \sqrt {2} b + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \, {\left (3^{\frac {3}{4}} 2^{\frac {3}{4}} a + 2 \, \sqrt {3} \sqrt {2} b + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \] Input:
integrate((c*x^2+b*x+a)/(3*x^4+2),x, algorithm="maxima")
Output:
1/48*3^(1/4)*2^(3/4)*(sqrt(3)*a - sqrt(2)*c)*log(sqrt(3)*x^2 + 3^(1/4)*2^( 3/4)*x + sqrt(2)) - 1/48*3^(1/4)*2^(3/4)*(sqrt(3)*a - sqrt(2)*c)*log(sqrt( 3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/24*(3^(3/4)*2^(3/4)*a - 2*sqrt(3 )*sqrt(2)*b + 2*3^(1/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/24*(3^(3/4)*2^(3/4)*a + 2*sqrt(3)*sqrt(2)*b + 2*3 ^(1/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4 )))
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (6^{\frac {3}{4}} a - 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \] Input:
integrate((c*x^2+b*x+a)/(3*x^4+2),x, algorithm="giac")
Output:
1/24*(6^(3/4)*a - 2*sqrt(6)*b + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4 )*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/24*(6^(3/4)*a + 2*sqrt(6)*b + 2*6^(1/4) *c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*(6^ (3/4)*a - 2*6^(1/4)*c)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48 *(6^(3/4)*a - 2*6^(1/4)*c)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))
Time = 6.03 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.14 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=\sum _{k=1}^4\ln \left (9\,a\,b^2-9\,a^2\,c-\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (864\,a-864\,b\,x\right )+144\,b\,c+x\,\left (108\,a^2-72\,c^2\right )\right )-6\,c^3+x\,\left (9\,b^3-18\,a\,b\,c\right )\right )\,\mathrm {root}\left (z^4+\frac {z^2\,\left (2304\,a\,c+1152\,b^2\right )}{55296}-\frac {z\,\left (288\,a^2\,b-192\,b\,c^2\right )}{55296}-\frac {a\,b^2\,c}{2304}+\frac {a^2\,c^2}{4608}+\frac {c^4}{13824}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right ) \] Input:
int((a + b*x + c*x^2)/(3*x^4 + 2),x)
Output:
symsum(log(9*a*b^2 - 9*a^2*c - root(z^4 + (z^2*(2304*a*c + 1152*b^2))/5529 6 - (z*(288*a^2*b - 192*b*c^2))/55296 - (a*b^2*c)/2304 + (a^2*c^2)/4608 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*(root(z^4 + (z^2*(2304*a*c + 1152*b ^2))/55296 - (z*(288*a^2*b - 192*b*c^2))/55296 - (a*b^2*c)/2304 + (a^2*c^2 )/4608 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*(864*a - 864*b*x) + 144*b* c + x*(108*a^2 - 72*c^2)) - 6*c^3 + x*(9*b^3 - 18*a*b*c))*root(z^4 + (z^2* (2304*a*c + 1152*b^2))/55296 - (z*(288*a^2*b - 192*b*c^2))/55296 - (a*b^2* c)/2304 + (a^2*c^2)/4608 + c^4/13824 + b^4/9216 + a^4/6144, z, k), k, 1, 4 )
Time = 0.16 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.13 \[ \int \frac {a+b x+c x^2}{2+3 x^4} \, dx=-\frac {\sqrt {6}\, 6^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}-2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) a}{24}-\frac {6^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}-2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) c}{12}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}-2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) b}{12}+\frac {\sqrt {6}\, 6^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}+2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) a}{24}+\frac {6^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}+2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) c}{12}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {\left (\sqrt {2}\, 6^{\frac {1}{4}}+2 \sqrt {3}\, x \right ) 6^{\frac {3}{4}}}{6 \sqrt {2}}\right ) b}{12}-\frac {\sqrt {6}\, 6^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {2}\, 6^{\frac {1}{4}} x +\sqrt {3}\, x^{2}+\sqrt {2}\right ) a}{48}+\frac {\sqrt {6}\, 6^{\frac {1}{4}} \mathrm {log}\left (\sqrt {2}\, 6^{\frac {1}{4}} x +\sqrt {3}\, x^{2}+\sqrt {2}\right ) a}{48}+\frac {6^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {2}\, 6^{\frac {1}{4}} x +\sqrt {3}\, x^{2}+\sqrt {2}\right ) c}{24}-\frac {6^{\frac {1}{4}} \mathrm {log}\left (\sqrt {2}\, 6^{\frac {1}{4}} x +\sqrt {3}\, x^{2}+\sqrt {2}\right ) c}{24} \] Input:
int((c*x^2+b*x+a)/(3*x^4+2),x)
Output:
( - 2*sqrt(6)*6**(1/4)*atan((sqrt(2)*6**(1/4) - 2*sqrt(3)*x)/(sqrt(2)*6**( 1/4)))*a - 4*6**(1/4)*atan((sqrt(2)*6**(1/4) - 2*sqrt(3)*x)/(sqrt(2)*6**(1 /4)))*c - 4*sqrt(6)*atan((sqrt(2)*6**(1/4) - 2*sqrt(3)*x)/(sqrt(2)*6**(1/4 )))*b + 2*sqrt(6)*6**(1/4)*atan((sqrt(2)*6**(1/4) + 2*sqrt(3)*x)/(sqrt(2)* 6**(1/4)))*a + 4*6**(1/4)*atan((sqrt(2)*6**(1/4) + 2*sqrt(3)*x)/(sqrt(2)*6 **(1/4)))*c - 4*sqrt(6)*atan((sqrt(2)*6**(1/4) + 2*sqrt(3)*x)/(sqrt(2)*6** (1/4)))*b - sqrt(6)*6**(1/4)*log( - sqrt(2)*6**(1/4)*x + sqrt(3)*x**2 + sq rt(2))*a + sqrt(6)*6**(1/4)*log(sqrt(2)*6**(1/4)*x + sqrt(3)*x**2 + sqrt(2 ))*a + 2*6**(1/4)*log( - sqrt(2)*6**(1/4)*x + sqrt(3)*x**2 + sqrt(2))*c - 2*6**(1/4)*log(sqrt(2)*6**(1/4)*x + sqrt(3)*x**2 + sqrt(2))*c)/48