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

Optimal. Leaf size=163 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

[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))

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Rubi [A]  time = 0.28505, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}+\frac{\left (\sqrt{6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 6^{3/4}}-\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac{\left (\sqrt{6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(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))

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Rubi in Sympy [A]  time = 30.0588, size = 144, normalized size = 0.88 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} - \frac{\sqrt [4]{6} \left (- \sqrt{6} a + 2 c\right ) \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{\sqrt [4]{6} \left (\sqrt{6} a + 2 c\right ) \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*b*atan(sqrt(6)*x**2/2)/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 + sq
rt(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

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Mathematica [A]  time = 0.165397, size = 129, normalized size = 0.79 \[ \frac{-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right ) \left (\sqrt{6} a+2 \left (\sqrt [4]{6} b+c\right )\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right ) \left (\sqrt{6} a-2 \sqrt [4]{6} b+2 c\right )-\left (\sqrt{6} a-2 c\right ) \left (\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )\right )}{8\ 6^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-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 + S
qrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(3/4))

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Maple [B]  time = 0.003, size = 241, normalized size = 1.5 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)))

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Maxima [A]  time = 1.53579, size = 252, normalized size = 1.55 \[ \frac{1}{48} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a - \sqrt{2} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{48} \cdot 3^{\frac{1}{4}} 2^{\frac{3}{4}}{\left (\sqrt{3} a - \sqrt{2} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \,{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} a - 2 \, \sqrt{3} \sqrt{2} b + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \,{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} a + 2 \, \sqrt{3} \sqrt{2} b + 2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)))

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 6.00409, size = 292, normalized size = 1.79 \[ \operatorname{RootSum}{\left (55296 t^{4} + t^{2} \left (2304 a c + 1152 b^{2}\right ) + t \left (- 288 a^{2} b + 192 b c^{2}\right ) + 9 a^{4} + 12 a^{2} c^{2} - 24 a b^{2} c + 6 b^{4} + 4 c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 13824 t^{3} a^{2} c + 27648 t^{3} a b^{2} + 9216 t^{3} c^{3} + 1728 t^{2} a^{3} b + 3456 t^{2} a b c^{2} - 2304 t^{2} b^{3} c + 216 t a^{5} - 576 t a^{3} c^{2} + 1296 t a^{2} b^{2} c + 288 t a b^{4} + 288 t a c^{4} + 288 t b^{2} c^{3} + 90 a^{4} b c - 90 a^{3} b^{3} + 60 a b^{3} c^{2} - 24 b^{5} c + 24 b c^{5}}{27 a^{6} - 18 a^{4} c^{2} + 144 a^{3} b^{2} c - 72 a^{2} b^{4} - 12 a^{2} c^{4} + 96 a b^{2} c^{3} - 48 b^{4} c^{2} + 8 c^{6}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + 1
296*_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))))

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GIAC/XCAS [A]  time = 0.227504, size = 193, normalized size = 1.18 \[ \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b + 2 \cdot 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 2 \cdot 6^{\frac{1}{4}} c\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + 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)*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))