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

Optimal. Leaf size=141 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (a*ArcTan[1 - 6^(1/4)*x])/(4*6^(1/4)) +
(a*ArcTan[1 + 6^(1/4)*x])/(4*6^(1/4)) - (a*Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqr
t[3]*x^2])/(8*6^(1/4)) + (a*Log[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(8*6
^(1/4))

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Rubi [A]  time = 0.227079, antiderivative size = 123, normalized size of antiderivative = 0.87, number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(2 + 3*x^4),x]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (a*ArcTan[1 - 6^(1/4)*x])/(4*6^(1/4)) +
(a*ArcTan[1 + 6^(1/4)*x])/(4*6^(1/4)) - (a*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*
6^(1/4)) + (a*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(1/4))

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Rubi in Sympy [A]  time = 23.5077, size = 110, normalized size = 0.78 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} + \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6**(3/4)*a*log(3*x**2 - 6**(3/4)*x + sqrt(6))/48 + 6**(3/4)*a*log(3*x**2 + 6**(
3/4)*x + sqrt(6))/48 + 6**(3/4)*a*atan(6**(1/4)*x - 1)/24 + 6**(3/4)*a*atan(6**(
1/4)*x + 1)/24 + sqrt(6)*b*atan(sqrt(6)*x**2/2)/12

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Mathematica [A]  time = 0.109898, size = 107, normalized size = 0.76 \[ \frac{-2 \left (\sqrt [4]{6} a+2 b\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt [4]{6} a-2 b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} a \left (\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )\right )}{8 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(6^(1/4)*a + 2*b)*ArcTan[1 - 6^(1/4)*x] + 2*(6^(1/4)*a - 2*b)*ArcTan[1 + 6^(
1/4)*x] + 6^(1/4)*a*(-Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + Log[2 + 2*6^(1/4)*x +
Sqrt[6]*x^2]))/(8*Sqrt[6])

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Maple [A]  time = 0.004, size = 129, normalized size = 0.9 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)

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Maxima [A]  time = 1.52984, size = 198, normalized size = 1.4 \[ \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a - 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*3^(3/4)*2^(3/4)*a*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/48*3^(
3/4)*2^(3/4)*a*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/24*sqrt(3)*(3^
(1/4)*2^(3/4)*a - 2*sqrt(2)*b)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)
*2^(3/4))) + 1/24*sqrt(3)*(3^(1/4)*2^(3/4)*a + 2*sqrt(2)*b)*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((b*x + a)/(3*x^4 + 2),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.796032, size = 88, normalized size = 0.62 \[ \operatorname{RootSum}{\left (18432 t^{4} + 384 t^{2} b^{2} - 96 t a^{2} b + 3 a^{4} + 2 b^{4}, \left ( t \mapsto t \log{\left (x + \frac{3072 t^{3} b^{2} + 192 t^{2} a^{2} b + 24 t a^{4} + 32 t b^{4} - 10 a^{2} b^{3}}{3 a^{5} - 8 a b^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(3*x**4+2),x)

[Out]

RootSum(18432*_t**4 + 384*_t**2*b**2 - 96*_t*a**2*b + 3*a**4 + 2*b**4, Lambda(_t
, _t*log(x + (3072*_t**3*b**2 + 192*_t**2*a**2*b + 24*_t*a**4 + 32*_t*b**4 - 10*
a**2*b**3)/(3*a**5 - 8*a*b**4))))

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GIAC/XCAS [A]  time = 0.222151, size = 155, normalized size = 1.1 \[ \frac{1}{48} \cdot 6^{\frac{3}{4}} a{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \cdot 6^{\frac{3}{4}} a{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*6^(3/4)*a*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*6^(3/4)*a*ln(x
^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/24*(6^(3/4)*a - 2*sqrt(6)*b)*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)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4)))