3.164 \(\int \frac{a+c x^2+d x^3}{2+3 x^4} \, dx\)

Optimal. Leaf size=154 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[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)) + (d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.283733, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2 + d*x^3)/(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)) + (d*Log[2 + 3*x^4])/12

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Rubi in Sympy [A]  time = 30.2491, size = 136, normalized size = 0.88 \[ \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} + \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} - \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c*x**2+a)/(3*x**4+2),x)

[Out]

d*log(3*x**4 + 2)/12 + 6**(1/4)*(-sqrt(6)*a + 2*c)*log(3*x**2 - 6**(3/4)*x + sqr
t(6))/48 - 6**(1/4)*(-sqrt(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)*a
tan(6**(1/4)*x + 1)/24

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Mathematica [A]  time = 0.275876, size = 148, normalized size = 0.96 \[ \frac{1}{48} \left (-\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+4 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a + 2*c
)*ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4)*x + Sqrt[6
]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[
2 + 3*x^4])/48

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Maple [B]  time = 0.003, size = 237, normalized size = 1.5 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+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)))+1/12*d*ln(3*x
^4+2)

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Maxima [A]  time = 1.55211, size = 263, normalized size = 1.71 \[ -\frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c - 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c*x^2 + a)/(3*x^4 + 2),x, algorithm="maxima")

[Out]

-1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c - 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(
3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c
+ 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1
/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/
4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3
/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4)))

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Fricas [A]  time = 0.412381, size = 4869, normalized size = 31.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c*x^2 + a)/(3*x^4 + 2),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*((sqrt(2)*(2*216^(1/4)*sqrt(6)*a*c*d - 216^(1/4)*sqrt(9*a^4 + 12*a^
2*c^2 + 4*c^4)*d)*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)) + 3*(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*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)) + (sqrt(
2)*(2*216^(1/4)*sqrt(6)*a*c*d - 216^(1/4)*sqrt(9*a^4 + 12*a^2*c^2 + 4*c^4)*d)*sq
rt((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))
- 3*(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*sqrt(6)*sqr
t(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 + 38
4*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)) + 12*(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)*(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 + 108
0*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)*sq
rt(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*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)) + 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)
)) + 12*(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)*arct
an(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)*(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*2
16^(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)*sq
rt((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)*sq
rt(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)))

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Sympy [A]  time = 2.01234, size = 148, normalized size = 0.96 \[ \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (6912 a c + 6912 d^{2}\right ) + t \left (- 1152 a c d - 384 d^{3}\right ) + 27 a^{4} + 36 a^{2} c^{2} + 48 a c d^{2} + 12 c^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 13824 t^{3} c + 3456 t^{2} c d + 216 t a^{3} - 432 t a c^{2} - 288 t c d^{2} - 18 a^{3} d + 36 a c^{2} d + 8 c d^{3}}{27 a^{4} - 12 c^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c*x**2+a)/(3*x**4+2),x)

[Out]

RootSum(165888*_t**4 - 55296*_t**3*d + _t**2*(6912*a*c + 6912*d**2) + _t*(-1152*
a*c*d - 384*d**3) + 27*a**4 + 36*a**2*c**2 + 48*a*c*d**2 + 12*c**4 + 8*d**4, Lam
bda(_t, _t*log(x + (-13824*_t**3*c + 3456*_t**2*c*d + 216*_t*a**3 - 432*_t*a*c**
2 - 288*_t*c*d**2 - 18*a**3*d + 36*a*c**2*d + 8*c*d**3)/(27*a**4 - 12*c**4))))

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GIAC/XCAS [A]  time = 0.226449, size = 185, normalized size = 1.2 \[ \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c + 4 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c - 4 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + 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 + 4*d)*ln(x^2 + sqrt(2)*
(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4)*c - 4*d)*ln(x^2 - sqrt(
2)*(2/3)^(1/4)*x + sqrt(2/3))