3.156 \(\int \frac{a+c x^2}{2+3 x^4} \, dx\)
Optimal. Leaf size=141 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]
[Out]
-((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a + 2*c)*ArcT
an[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)) + ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3
/4))
_______________________________________________________________________________________
Rubi [A] time = 0.233549, antiderivative size = 141, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353
\[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(2 + 3*x^4),x]
[Out]
-((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a + 2*c)*ArcT
an[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)) + ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3
/4))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.9858, size = 124, normalized size = 0.88 \[ \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} - \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(3*x**4+2),x)
[Out]
6**(1/4)*(-sqrt(6)*a + 2*c)*log(3*x**2 - 6**(3/4)*x + sqrt(6))/48 - 6**(1/4)*(-s
qrt(6)*a + 2*c)*log(3*x**2 + 6**(3/4)*x + sqrt(6))/48 + 6**(1/4)*(sqrt(6)*a + 2*
c)*atan(6**(1/4)*x - 1)/24 + 6**(1/4)*(sqrt(6)*a + 2*c)*atan(6**(1/4)*x + 1)/24
_______________________________________________________________________________________
Mathematica [A] time = 0.0924101, size = 113, normalized size = 0.8 \[ \frac{-\left (\sqrt{6} a-2 c\right ) \left (\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )\right )-2 \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{8\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(2 + 3*x^4),x]
[Out]
(-2*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*(Sqrt[6]*a + 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))
_______________________________________________________________________________________
Maple [B] time = 0.003, size = 226, normalized size = 1.6 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(3*x^4+2),x)
[Out]
1/24*a*3^(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*a*3^
(1/2)*6^(1/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/48*a*3^(1/2)*6^(
1/4)*2^(1/2)*ln((x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2))/(x^2-1/3*3^(1/2)
*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))+1/72*c*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/
2)*3^(1/2)*6^(3/4)*x+1)+1/72*c*3^(1/2)*6^(3/4)*2^(1/2)*arctan(1/6*2^(1/2)*3^(1/2
)*6^(3/4)*x-1)+1/144*c*3^(1/2)*6^(3/4)*2^(1/2)*ln((x^2-1/3*3^(1/2)*6^(1/4)*x*2^(
1/2)+1/3*6^(1/2))/(x^2+1/3*3^(1/2)*6^(1/4)*x*2^(1/2)+1/3*6^(1/2)))
_______________________________________________________________________________________
Maxima [A] time = 1.53162, size = 225, normalized size = 1.6 \[ \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a + \sqrt{2} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a + \sqrt{2} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{48} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a - \sqrt{2} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{48} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a - \sqrt{2} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(3*x^4 + 2),x, algorithm="maxima")
[Out]
1/24*3^(1/4)*2^(3/4)*(sqrt(3)*a + sqrt(2)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(
3)*x + 3^(1/4)*2^(3/4))) + 1/24*3^(1/4)*2^(3/4)*(sqrt(3)*a + sqrt(2)*c)*arctan(1
/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 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))
_______________________________________________________________________________________
Fricas [A] time = 0.409903, size = 4479, normalized size = 31.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
1/8*sqrt(2)*((9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4)*(2*sqrt(6)*a*c - sqrt(9*a^4 + 12
*a^2*c^2 + 4*c^4))*log(-6*sqrt(6)*(243*a^10 + 2754*a^8*c^2 + 4968*a^6*c^4 + 3312
*a^4*c^6 + 816*a^2*c^8 + 32*c^10)*x^2 + 216*(27*a^7*c + 78*a^5*c^3 + 52*a^3*c^5
+ 8*a*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*x^2 + 2*sqrt(2)*(9*a^4 + 12*a^2*c^2
+ 4*c^4)^(1/4)*(216^(1/4)*sqrt(6)*(189*a^6*c + 630*a^4*c^3 + 252*a^2*c^5 + 8*c^7
)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*x - 3*216^(1/4)*(81*a^9 + 1188*a^7*c^2 + 2016
*a^5*c^4 + 1008*a^3*c^6 + 112*a*c^8)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqr
t(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(
6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)) - 12*(81*a^8 + 864*a^6*c^2 + 1080*a^4*
c^4 + 384*a^2*c^6 + 16*c^8 - 6*sqrt(6)*(9*a^5*c + 20*a^3*c^3 + 4*a*c^5)*sqrt(9*a
^4 + 12*a^2*c^2 + 4*c^4))*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)) - (9*a^4 + 12*a^2*c^
2 + 4*c^4)^(1/4)*(2*sqrt(6)*a*c - sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))*log(-6*sqrt(
6)*(243*a^10 + 2754*a^8*c^2 + 4968*a^6*c^4 + 3312*a^4*c^6 + 816*a^2*c^8 + 32*c^1
0)*x^2 + 216*(27*a^7*c + 78*a^5*c^3 + 52*a^3*c^5 + 8*a*c^7)*sqrt(9*a^4 + 12*a^2*
c^2 + 4*c^4)*x^2 - 2*sqrt(2)*(9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4)*(216^(1/4)*sqrt(
6)*(189*a^6*c + 630*a^4*c^3 + 252*a^2*c^5 + 8*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c
^4)*x - 3*216^(1/4)*(81*a^9 + 1188*a^7*c^2 + 2016*a^5*c^4 + 1008*a^3*c^6 + 112*a
*c^8)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 +
4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*
c^4)*a*c)) - 12*(81*a^8 + 864*a^6*c^2 + 1080*a^4*c^4 + 384*a^2*c^6 + 16*c^8 - 6*
sqrt(6)*(9*a^5*c + 20*a^3*c^3 + 4*a*c^5)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))*sqrt(
9*a^4 + 12*a^2*c^2 + 4*c^4)) + 4*(9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4)*sqrt(9*a^4 -
12*a^2*c^2 + 4*c^4)*arctan(6*(9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4)*(sqrt(6)*sqrt(9
*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^3 + 2*a*c^2) - 2*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4
)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*c)/(sqrt(2)*sqrt(1/3)*(216^(1/4)*sqrt(6)*sqrt
(9*a^4 + 12*a^2*c^2 + 4*c^4)*(9*a^4 - 4*c^4) - 12*216^(1/4)*(9*a^5*c - 4*a*c^5))
*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a
*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c
))*sqrt((3*sqrt(6)*(243*a^10 + 2754*a^8*c^2 + 4968*a^6*c^4 + 3312*a^4*c^6 + 816*
a^2*c^8 + 32*c^10)*x^2 - 108*(27*a^7*c + 78*a^5*c^3 + 52*a^3*c^5 + 8*a*c^7)*sqrt
(9*a^4 + 12*a^2*c^2 + 4*c^4)*x^2 + sqrt(2)*(9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4)*(2
16^(1/4)*sqrt(6)*(189*a^6*c + 630*a^4*c^3 + 252*a^2*c^5 + 8*c^7)*sqrt(9*a^4 + 12
*a^2*c^2 + 4*c^4)*x - 3*216^(1/4)*(81*a^9 + 1188*a^7*c^2 + 2016*a^5*c^4 + 1008*a
^3*c^6 + 112*a*c^8)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt(9*a^4 +
12*a^2*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9*a^4 + 1
2*a^2*c^2 + 4*c^4)*a*c)) + 6*(81*a^8 + 864*a^6*c^2 + 1080*a^4*c^4 + 384*a^2*c^6
+ 16*c^8 - 6*sqrt(6)*(9*a^5*c + 20*a^3*c^3 + 4*a*c^5)*sqrt(9*a^4 + 12*a^2*c^2 +
4*c^4))*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))/(sqrt(6)*(243*a^10 + 2754*a^8*c^2 + 49
68*a^6*c^4 + 3312*a^4*c^6 + 816*a^2*c^8 + 32*c^10) - 36*(27*a^7*c + 78*a^5*c^3 +
52*a^3*c^5 + 8*a*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))) + sqrt(2)*(216^(1/4)*s
qrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*(9*a^4 - 4*c^4)*x - 12*216^(1/4)*(9*a^5*
c - 4*a*c^5)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt(9*a^4 + 12*a^2
*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9*a^4 + 12*a^2*c
^2 + 4*c^4)*a*c)) + 6*(18*a^4*c - 8*c^5 - sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^
4)*(3*a^3 - 2*a*c^2))*(9*a^4 + 12*a^2*c^2 + 4*c^4)^(1/4))) + 4*(9*a^4 + 12*a^2*c
^2 + 4*c^4)^(1/4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*arctan(6*(9*a^4 + 12*a^2*c^2
+ 4*c^4)^(1/4)*(sqrt(6)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*(3*a^3 + 2*a*c^2) - 2*s
qrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*sqrt(9*a^4 - 12*a^2*c^2 + 4*c^4)*c)/(sqrt(2)*sqr
t(1/3)*(216^(1/4)*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*(9*a^4 - 4*c^4) - 12*
216^(1/4)*(9*a^5*c - 4*a*c^5))*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt
(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9
*a^4 + 12*a^2*c^2 + 4*c^4)*a*c))*sqrt((3*sqrt(6)*(243*a^10 + 2754*a^8*c^2 + 4968
*a^6*c^4 + 3312*a^4*c^6 + 816*a^2*c^8 + 32*c^10)*x^2 - 108*(27*a^7*c + 78*a^5*c^
3 + 52*a^3*c^5 + 8*a*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*x^2 - sqrt(2)*(9*a^4
+ 12*a^2*c^2 + 4*c^4)^(1/4)*(216^(1/4)*sqrt(6)*(189*a^6*c + 630*a^4*c^3 + 252*a^
2*c^5 + 8*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*x - 3*216^(1/4)*(81*a^9 + 1188*a
^7*c^2 + 2016*a^5*c^4 + 1008*a^3*c^6 + 112*a*c^8)*x)*sqrt((9*a^4 + 12*a^2*c^2 +
4*c^4 - 2*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*
c^4 - 4*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)) + 6*(81*a^8 + 864*a^6*c^2
+ 1080*a^4*c^4 + 384*a^2*c^6 + 16*c^8 - 6*sqrt(6)*(9*a^5*c + 20*a^3*c^3 + 4*a*c
^5)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4))/(sqrt(6)
*(243*a^10 + 2754*a^8*c^2 + 4968*a^6*c^4 + 3312*a^4*c^6 + 816*a^2*c^8 + 32*c^10)
- 36*(27*a^7*c + 78*a^5*c^3 + 52*a^3*c^5 + 8*a*c^7)*sqrt(9*a^4 + 12*a^2*c^2 + 4
*c^4))) + sqrt(2)*(216^(1/4)*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*(9*a^4 - 4
*c^4)*x - 12*216^(1/4)*(9*a^5*c - 4*a*c^5)*x)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 -
2*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4
*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*c)) - 6*(18*a^4*c - 8*c^5 - sqrt(6)*
sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*(3*a^3 - 2*a*c^2))*(9*a^4 + 12*a^2*c^2 + 4*c^4)
^(1/4))))/((2*216^(1/4)*sqrt(6)*a*c - 216^(1/4)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)
)*sqrt((9*a^4 + 12*a^2*c^2 + 4*c^4 - 2*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*
a*c)/(9*a^4 + 36*a^2*c^2 + 4*c^4 - 4*sqrt(6)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*a*
c)))
_______________________________________________________________________________________
Sympy [A] time = 0.610627, size = 68, normalized size = 0.48 \[ \operatorname{RootSum}{\left (55296 t^{4} + 2304 t^{2} a c + 9 a^{4} + 12 a^{2} c^{2} + 4 c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 4608 t^{3} c + 72 t a^{3} - 144 t a c^{2}}{9 a^{4} - 4 c^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(3*x**4+2),x)
[Out]
RootSum(55296*_t**4 + 2304*_t**2*a*c + 9*a**4 + 12*a**2*c**2 + 4*c**4, Lambda(_t
, _t*log(x + (-4608*_t**3*c + 72*_t*a**3 - 144*_t*a*c**2)/(9*a**4 - 4*c**4))))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.225005, size = 177, normalized size = 1.26 \[ \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(3*x^4 + 2),x, algorithm="giac")
[Out]
1/24*(6^(3/4)*a + 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*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)*ln(x^2 + sqrt(2)*(2/3)^
(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4)*c)*ln(x^2 - sqrt(2)*(2/3)^(1/
4)*x + sqrt(2/3))