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

Optimal. Leaf size=132 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

-(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 + Sqrt[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)) + (d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.208991, antiderivative size = 114, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + d*x^3)/(2 + 3*x^4),x]

[Out]

-(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)) + (d*Log[2 + 3*x^4])/12

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Rubi in Sympy [A]  time = 23.7964, size = 102, normalized size = 0.77 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+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 + d*log(3*x**4 + 2)/12

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Mathematica [A]  time = 0.0610137, size = 108, normalized size = 0.82 \[ \frac{1}{48} \left (-6^{3/4} a \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} a \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+4 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*6^(3/4)*a*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*a*ArcTan[1 + 6^(1/4)*x] - 6^(3/4
)*a*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*a*Log[2 + 2*6^(1/4)*x + Sqrt[6]
*x^2] + 4*d*Log[2 + 3*x^4])/48

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Maple [A]  time = 0.004, size = 125, normalized size = 1. \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+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*d*ln(3*x^4+2)

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Maxima [A]  time = 1.54513, size = 201, normalized size = 1.52 \[ \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d + 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*3^(3/4)*2^(3/4)*a*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*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^
(3/4))) + 1/144*3^(3/4)*2^(3/4)*(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)*(2*3^(1/4)*2^(1/4)*d - 3*a)*l
og(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

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Fricas [A]  time = 0.246206, size = 389, normalized size = 2.95 \[ \frac{1}{576} \cdot 24^{\frac{3}{4}}{\left ({\left (2 \cdot 24^{\frac{1}{4}} d + 3 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}\right )} \log \left (2 \, \sqrt{6} a^{2} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right ) +{\left (2 \cdot 24^{\frac{1}{4}} d - 3 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}\right )} \log \left (2 \, \sqrt{6} a^{2} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right ) - 12 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} + 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right ) - 12 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} - 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/576*24^(3/4)*((2*24^(1/4)*d + 3*sqrt(2)*(a^4)^(1/4))*log(2*sqrt(6)*a^2*x^2 + 2
*24^(1/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(a^4)) + (2*24^(1/4)*d - 3*sqrt(2)*(a^
4)^(1/4))*log(2*sqrt(6)*a^2*x^2 - 2*24^(1/4)*sqrt(2)*(a^4)^(1/4)*a*x + 4*sqrt(a^
4)) - 12*sqrt(2)*(a^4)^(1/4)*arctan(sqrt(2)*(a^4)^(1/4)/(24^(1/4)*a*x + 24^(1/4)
*sqrt(1/6)*a*sqrt(sqrt(6)*(sqrt(6)*a^2*x^2 + 24^(1/4)*sqrt(2)*(a^4)^(1/4)*a*x +
2*sqrt(a^4))/a^2) + sqrt(2)*(a^4)^(1/4))) - 12*sqrt(2)*(a^4)^(1/4)*arctan(sqrt(2
)*(a^4)^(1/4)/(24^(1/4)*a*x + 24^(1/4)*sqrt(1/6)*a*sqrt(sqrt(6)*(sqrt(6)*a^2*x^2
- 24^(1/4)*sqrt(2)*(a^4)^(1/4)*a*x + 2*sqrt(a^4))/a^2) - sqrt(2)*(a^4)^(1/4))))

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Sympy [A]  time = 0.503587, size = 51, normalized size = 0.39 \[ \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + 6912 t^{2} d^{2} - 384 t d^{3} + 27 a^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{24 t - 2 d}{3 a} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(165888*_t**4 - 55296*_t**3*d + 6912*_t**2*d**2 - 384*_t*d**3 + 27*a**4 +
8*d**4, Lambda(_t, _t*log(x + (24*_t - 2*d)/(3*a))))

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GIAC/XCAS [A]  time = 0.223524, size = 147, normalized size = 1.11 \[ \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a + 4 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 4 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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