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

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

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (c*ArcTan[1 - 6^(1/4)*x])/(2*6^(3/4)) +
(c*ArcTan[1 + 6^(1/4)*x])/(2*6^(3/4)) + (c*Log[Sqrt[2] - 2^(3/4)*3^(1/4)*x + Sqr
t[3]*x^2])/(4*6^(3/4)) - (c*Log[Sqrt[2] + 2^(3/4)*3^(1/4)*x + Sqrt[3]*x^2])/(4*6
^(3/4)) + (d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.323716, antiderivative size = 136, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - (c*ArcTan[1 - 6^(1/4)*x])/(2*6^(3/4)) +
(c*ArcTan[1 + 6^(1/4)*x])/(2*6^(3/4)) + (c*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(4*
6^(3/4)) - (c*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(4*6^(3/4)) + (d*Log[2 + 3*x^4])
/12

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Rubi in Sympy [A]  time = 36.8169, size = 122, normalized size = 0.79 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{\sqrt [4]{6} c \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} c \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.126703, size = 125, normalized size = 0.81 \[ \frac{1}{24} \left (-2 \sqrt [4]{6} \left (\sqrt [4]{6} b+c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (c-\sqrt [4]{6} b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+2 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*6^(1/4)*(6^(1/4)*b + c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(-(6^(1/4)*b) + c)
*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*
c*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/24

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Maple [A]  time = 0.005, size = 140, normalized size = 0.9 \[{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)

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Maxima [A]  time = 1.61724, size = 235, normalized size = 1.53 \[ \frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c - 6 \, b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c + 6 \, b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d - \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d + \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/72*sqrt(3)*sqrt(2)*(3^(3/4)*2^(3/4)*c - 6*b)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqr
t(3)*x + 3^(1/4)*2^(3/4))) + 1/72*sqrt(3)*sqrt(2)*(3^(3/4)*2^(3/4)*c + 6*b)*arct
an(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/72*3^(3/4)*2^(1/4)*(
3^(1/4)*2^(3/4)*d - sqrt(3)*c)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) +
1/72*3^(3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d + sqrt(3)*c)*log(sqrt(3)*x^2 - 3^(1/4)*2
^(3/4)*x + sqrt(2))

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.24829, size = 189, normalized size = 1.23 \[ \operatorname{RootSum}{\left (82944 t^{4} - 27648 t^{3} d + t^{2} \left (1728 b^{2} + 3456 d^{2}\right ) + t \left (- 288 b^{2} d + 288 b c^{2} - 192 d^{3}\right ) + 9 b^{4} + 12 b^{2} d^{2} - 24 b c^{2} d + 6 c^{4} + 4 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 3456 t^{3} c^{2} + 864 t^{2} b^{3} + 864 t^{2} c^{2} d - 144 t b^{3} d - 108 t b^{2} c^{2} - 72 t c^{2} d^{2} + 9 b^{5} + 6 b^{3} d^{2} + 9 b^{2} c^{2} d - 9 b c^{4} + 2 c^{2} d^{3}}{18 b^{4} c - 3 c^{5}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(82944*_t**4 - 27648*_t**3*d + _t**2*(1728*b**2 + 3456*d**2) + _t*(-288*b
**2*d + 288*b*c**2 - 192*d**3) + 9*b**4 + 12*b**2*d**2 - 24*b*c**2*d + 6*c**4 +
4*d**4, Lambda(_t, _t*log(x + (-3456*_t**3*c**2 + 864*_t**2*b**3 + 864*_t**2*c**
2*d - 144*_t*b**3*d - 108*_t*b**2*c**2 - 72*_t*c**2*d**2 + 9*b**5 + 6*b**3*d**2
+ 9*b**2*c**2*d - 9*b*c**4 + 2*c**2*d**3)/(18*b**4*c - 3*c**5))))

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GIAC/XCAS [A]  time = 0.225389, size = 167, normalized size = 1.08 \[ -\frac{1}{12} \,{\left (\sqrt{6} b - 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \,{\left (\sqrt{6} b + 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \,{\left (6^{\frac{1}{4}} c - 2 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{1}{4}} c + 2 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(sqrt(6)*b - 6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3
)^(1/4))) + 1/12*(sqrt(6)*b + 6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - s
qrt(2)*(2/3)^(1/4))) - 1/24*(6^(1/4)*c - 2*d)*ln(x^2 + sqrt(2)*(2/3)^(1/4)*x + s
qrt(2/3)) + 1/24*(6^(1/4)*c + 2*d)*ln(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))