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

Optimal. Leaf size=132 \[ \frac{c \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

-(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 + Sqrt[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 = 0.23999, antiderivative size = 114, normalized size of antiderivative = 0.86, number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(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 = 28.3706, size = 102, normalized size = 0.77 \[ \frac{\sqrt [4]{6} c \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} c \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 = 0.0488684, size = 108, normalized size = 0.82 \[ \frac{1}{24} \left (\sqrt [4]{6} c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+2 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*6^(1/4)*c*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*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 = 0.003, size = 125, normalized size = 1. \[{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c*x^2)/(3*x^4+2),x)

[Out]

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 = 1.53629, size = 205, normalized size = 1.55 \[ \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d - \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d + \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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(s
qrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/12*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))) + 1/12*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 [A]  time = 0.2405, size = 440, normalized size = 3.33 \[ \frac{1}{1296} \cdot 54^{\frac{3}{4}}{\left ({\left (2 \cdot 54^{\frac{1}{4}} d - 3 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}}\right )} \log \left (3 \, \sqrt{6} c^{3} x^{2} + 54^{\frac{1}{4}} \sqrt{6} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 6 \, \sqrt{c^{4}} c\right ) +{\left (2 \cdot 54^{\frac{1}{4}} d + 3 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}}\right )} \log \left (3 \, \sqrt{6} c^{3} x^{2} - 54^{\frac{1}{4}} \sqrt{6} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 6 \, \sqrt{c^{4}} c\right ) - 12 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{9 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}{3 \cdot 54^{\frac{1}{4}} \sqrt{6} c^{3} x + 54^{\frac{1}{4}} \sqrt{6} \sqrt{\frac{1}{2}} c^{3} \sqrt{\frac{\sqrt{6}{\left (3 \, \sqrt{6} c^{3} x^{2} + 54^{\frac{1}{4}} \sqrt{6} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 6 \, \sqrt{c^{4}} c\right )}}{c^{3}}} + 9 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}\right ) - 12 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{9 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}{3 \cdot 54^{\frac{1}{4}} \sqrt{6} c^{3} x + 54^{\frac{1}{4}} \sqrt{6} \sqrt{\frac{1}{2}} c^{3} \sqrt{\frac{\sqrt{6}{\left (3 \, \sqrt{6} c^{3} x^{2} - 54^{\frac{1}{4}} \sqrt{6} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 6 \, \sqrt{c^{4}} c\right )}}{c^{3}}} - 9 \, \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1296*54^(3/4)*((2*54^(1/4)*d - 3*sqrt(2)*(c^4)^(1/4))*log(3*sqrt(6)*c^3*x^2 +
54^(1/4)*sqrt(6)*sqrt(2)*(c^4)^(3/4)*x + 6*sqrt(c^4)*c) + (2*54^(1/4)*d + 3*sqrt
(2)*(c^4)^(1/4))*log(3*sqrt(6)*c^3*x^2 - 54^(1/4)*sqrt(6)*sqrt(2)*(c^4)^(3/4)*x
+ 6*sqrt(c^4)*c) - 12*sqrt(2)*(c^4)^(1/4)*arctan(9*sqrt(2)*(c^4)^(3/4)/(3*54^(1/
4)*sqrt(6)*c^3*x + 54^(1/4)*sqrt(6)*sqrt(1/2)*c^3*sqrt(sqrt(6)*(3*sqrt(6)*c^3*x^
2 + 54^(1/4)*sqrt(6)*sqrt(2)*(c^4)^(3/4)*x + 6*sqrt(c^4)*c)/c^3) + 9*sqrt(2)*(c^
4)^(3/4))) - 12*sqrt(2)*(c^4)^(1/4)*arctan(9*sqrt(2)*(c^4)^(3/4)/(3*54^(1/4)*sqr
t(6)*c^3*x + 54^(1/4)*sqrt(6)*sqrt(1/2)*c^3*sqrt(sqrt(6)*(3*sqrt(6)*c^3*x^2 - 54
^(1/4)*sqrt(6)*sqrt(2)*(c^4)^(3/4)*x + 6*sqrt(c^4)*c)/c^3) - 9*sqrt(2)*(c^4)^(3/
4))))

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Sympy [A]  time = 0.428352, size = 70, normalized size = 0.53 \[ \operatorname{RootSum}{\left (41472 t^{4} - 13824 t^{3} d + 1728 t^{2} d^{2} - 96 t d^{3} + 3 c^{4} + 2 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{3456 t^{3} - 864 t^{2} d + 72 t d^{2} - 2 d^{3}}{3 c^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(41472*_t**4 - 13824*_t**3*d + 1728*_t**2*d**2 - 96*_t*d**3 + 3*c**4 + 2*
d**4, Lambda(_t, _t*log(x + (3456*_t**3 - 864*_t**2*d + 72*_t*d**2 - 2*d**3)/(3*
c**3))))

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GIAC/XCAS [A]  time = 0.223075, size = 147, normalized size = 1.11 \[ \frac{1}{12} \cdot 6^{\frac{1}{4}} c \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \cdot 6^{\frac{1}{4}} c \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \,{\left (6^{\frac{1}{4}} c - 2 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{1}{4}} c + 2 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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