3.30 \(\int \frac{a^{2/3} C+2 C x^2}{a+8 x^3} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0651268, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{4} C \log \left (\sqrt [3]{a}+2 x\right )-\frac{C \tan ^{-1}\left (\frac{\sqrt [3]{a}-4 x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(a^(2/3)*C + 2*C*x^2)/(a + 8*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.77557, size = 44, normalized size = 0.94 \[ \frac{C \log{\left (\sqrt [3]{a} + 2 x \right )}}{4} - \frac{\sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{4 x}{3}\right )}{\sqrt [3]{a}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**(2/3)*C+2*C*x**2)/(8*x**3+a),x)

[Out]

C*log(a**(1/3) + 2*x)/4 - sqrt(3)*C*atan(sqrt(3)*(a**(1/3)/3 - 4*x/3)/a**(1/3))/
6

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Mathematica [A]  time = 0.0459204, size = 72, normalized size = 1.53 \[ \frac{1}{12} C \left (-\log \left (a^{2/3}-2 \sqrt [3]{a} x+4 x^2\right )+\log \left (a+8 x^3\right )+2 \log \left (\sqrt [3]{a}+2 x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^(2/3)*C + 2*C*x^2)/(a + 8*x^3),x]

[Out]

(C*(-2*Sqrt[3]*ArcTan[(1 - (4*x)/a^(1/3))/Sqrt[3]] + 2*Log[a^(1/3) + 2*x] - Log[
a^(2/3) - 2*a^(1/3)*x + 4*x^2] + Log[a + 8*x^3]))/12

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Maple [B]  time = 0.009, size = 84, normalized size = 1.8 \[{\frac{C{8}^{{\frac{2}{3}}}}{24}\ln \left ( x+{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ) }-{\frac{C{8}^{{\frac{2}{3}}}}{48}\ln \left ({x}^{2}-{\frac{x{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ) }+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}-1 \right ) } \right ) }+{\frac{C\ln \left ( 8\,{x}^{3}+a \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^(2/3)*C+2*C*x^2)/(8*x^3+a),x)

[Out]

1/24*C*8^(2/3)*ln(x+1/8*8^(2/3)*a^(1/3))-1/48*C*8^(2/3)*ln(x^2-1/8*x*8^(2/3)*a^(
1/3)+1/8*8^(1/3)*a^(2/3))+1/24*C*8^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*8^(1/3)/a
^(1/3)*x-1))+1/12*C*ln(8*x^3+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232747, size = 61, normalized size = 1.3 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} C \log \left (2 \, a^{\frac{2}{3}} x + a\right ) + 2 \, C \arctan \left (\frac{4 \, \sqrt{3} a^{\frac{2}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*(sqrt(3)*C*log(2*a^(2/3)*x + a) + 2*C*arctan(1/3*(4*sqrt(3)*a^(2/3)
*x - sqrt(3)*a)/a))

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Sympy [A]  time = 0.807761, size = 85, normalized size = 1.81 \[ C \left (\frac{\log{\left (\frac{\sqrt [3]{a}}{2} + x \right )}}{4} - \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} - \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12} + \frac{\sqrt{3} i \log{\left (x + \frac{- C \sqrt [3]{a} + \sqrt{3} i C \sqrt [3]{a}}{4 C} \right )}}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**(2/3)*C+2*C*x**2)/(8*x**3+a),x)

[Out]

C*(log(a**(1/3)/2 + x)/4 - sqrt(3)*I*log(x + (-C*a**(1/3) - sqrt(3)*I*C*a**(1/3)
)/(4*C))/12 + sqrt(3)*I*log(x + (-C*a**(1/3) + sqrt(3)*I*C*a**(1/3))/(4*C))/12)

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GIAC/XCAS [A]  time = 0.243111, size = 147, normalized size = 3.13 \[ \frac{\sqrt{3}{\left (\sqrt{3} a i + a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + \left (-a\right )^{\frac{1}{3}}\right )}}{3 \, \left (-a\right )^{\frac{1}{3}}}\right )}{12 \, a} - \frac{{\left (\sqrt{3} a i - 3 \, a\right )} C{\rm ln}\left (x^{2} + \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} x + \frac{1}{4} \, \left (-a\right )^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (C \left (-a\right )^{\frac{2}{3}} + 2 \, C a^{\frac{2}{3}}\right )} \left (-a\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \frac{1}{2} \, \left (-a\right )^{\frac{1}{3}} \right |}\right )}{12 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*C*x^2 + C*a^(2/3))/(8*x^3 + a),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(sqrt(3)*a*i + a)*C*arctan(1/3*sqrt(3)*(4*x + (-a)^(1/3))/(-a)^(1/3
))/a - 1/24*(sqrt(3)*a*i - 3*a)*C*ln(x^2 + 1/2*(-a)^(1/3)*x + 1/4*(-a)^(2/3))/a
- 1/12*(C*(-a)^(2/3) + 2*C*a^(2/3))*(-a)^(1/3)*ln(abs(x - 1/2*(-a)^(1/3)))/a