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

Optimal. Leaf size=176 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

(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)) + ((Sqrt[6]*a - 2*c)*Lo
g[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12

_______________________________________________________________________________________

Rubi [A]  time = 0.339016, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(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)) + ((Sqrt[6]*a - 2*c)*Lo
g[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.9786, size = 156, normalized size = 0.89 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} + \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} - \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*b*atan(sqrt(6)*x**2/2)/12 + d*log(3*x**4 + 2)/12 + 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)*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 = 0.259358, size = 164, normalized size = 0.93 \[ \frac{1}{48} \left (-2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right ) \left (\sqrt{6} a+2 \left (\sqrt [4]{6} b+c\right )\right )+2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right ) \left (\sqrt{6} a-2 \sqrt [4]{6} b+2 c\right )-\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\sqrt [4]{6} \left (\sqrt{6} a-2 c\right ) \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+4 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*6^(1/4)*(Sqrt[6]*a + 2*(6^(1/4)*b + c))*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(S
qrt[6]*a - 2*6^(1/4)*b + 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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 252, normalized size = 1.4 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c*x^2+b*x+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)))+1/12*b*arctan(1/2*x^2*6^(1/2))*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)

_______________________________________________________________________________________

Maxima [A]  time = 1.5612, size = 279, normalized size = 1.59 \[ -\frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c - 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} \sqrt{2} c + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c - 6 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \, \sqrt{3}{\left (3 \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} c + 6 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 12.9611, size = 580, normalized size = 3.3 \[ \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (6912 a c + 3456 b^{2} + 6912 d^{2}\right ) + t \left (- 864 a^{2} b - 1152 a c d - 576 b^{2} d + 576 b c^{2} - 384 d^{3}\right ) + 27 a^{4} + 72 a^{2} b d + 36 a^{2} c^{2} - 72 a b^{2} c + 48 a c d^{2} + 18 b^{4} + 24 b^{2} d^{2} - 48 b c^{2} d + 12 c^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 41472 t^{3} a^{2} c + 82944 t^{3} a b^{2} + 27648 t^{3} c^{3} + 5184 t^{2} a^{3} b + 10368 t^{2} a^{2} c d - 20736 t^{2} a b^{2} d + 10368 t^{2} a b c^{2} - 6912 t^{2} b^{3} c - 6912 t^{2} c^{3} d + 648 t a^{5} - 864 t a^{3} b d - 1728 t a^{3} c^{2} + 3888 t a^{2} b^{2} c - 864 t a^{2} c d^{2} + 864 t a b^{4} + 1728 t a b^{2} d^{2} - 1728 t a b c^{2} d + 864 t a c^{4} + 1152 t b^{3} c d + 864 t b^{2} c^{3} + 576 t c^{3} d^{2} - 54 a^{5} d + 270 a^{4} b c - 270 a^{3} b^{3} + 36 a^{3} b d^{2} + 144 a^{3} c^{2} d - 324 a^{2} b^{2} c d + 24 a^{2} c d^{3} - 72 a b^{4} d + 180 a b^{3} c^{2} - 48 a b^{2} d^{3} + 72 a b c^{2} d^{2} - 72 a c^{4} d - 72 b^{5} c - 48 b^{3} c d^{2} - 72 b^{2} c^{3} d + 72 b c^{5} - 16 c^{3} d^{3}}{81 a^{6} - 54 a^{4} c^{2} + 432 a^{3} b^{2} c - 216 a^{2} b^{4} - 36 a^{2} c^{4} + 288 a b^{2} c^{3} - 144 b^{4} c^{2} + 24 c^{6}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(165888*_t**4 - 55296*_t**3*d + _t**2*(6912*a*c + 3456*b**2 + 6912*d**2)
+ _t*(-864*a**2*b - 1152*a*c*d - 576*b**2*d + 576*b*c**2 - 384*d**3) + 27*a**4 +
72*a**2*b*d + 36*a**2*c**2 - 72*a*b**2*c + 48*a*c*d**2 + 18*b**4 + 24*b**2*d**2
- 48*b*c**2*d + 12*c**4 + 8*d**4, Lambda(_t, _t*log(x + (-41472*_t**3*a**2*c +
82944*_t**3*a*b**2 + 27648*_t**3*c**3 + 5184*_t**2*a**3*b + 10368*_t**2*a**2*c*d
- 20736*_t**2*a*b**2*d + 10368*_t**2*a*b*c**2 - 6912*_t**2*b**3*c - 6912*_t**2*
c**3*d + 648*_t*a**5 - 864*_t*a**3*b*d - 1728*_t*a**3*c**2 + 3888*_t*a**2*b**2*c
- 864*_t*a**2*c*d**2 + 864*_t*a*b**4 + 1728*_t*a*b**2*d**2 - 1728*_t*a*b*c**2*d
+ 864*_t*a*c**4 + 1152*_t*b**3*c*d + 864*_t*b**2*c**3 + 576*_t*c**3*d**2 - 54*a
**5*d + 270*a**4*b*c - 270*a**3*b**3 + 36*a**3*b*d**2 + 144*a**3*c**2*d - 324*a*
*2*b**2*c*d + 24*a**2*c*d**3 - 72*a*b**4*d + 180*a*b**3*c**2 - 48*a*b**2*d**3 +
72*a*b*c**2*d**2 - 72*a*c**4*d - 72*b**5*c - 48*b**3*c*d**2 - 72*b**2*c**3*d + 7
2*b*c**5 - 16*c**3*d**3)/(81*a**6 - 54*a**4*c**2 + 432*a**3*b**2*c - 216*a**2*b*
*4 - 36*a**2*c**4 + 288*a*b**2*c**3 - 144*b**4*c**2 + 24*c**6))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23005, size = 201, normalized size = 1.14 \[ \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c + 4 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c - 4 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + 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))