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

Optimal. Leaf size=48 \[ \frac{C \log \left (\sqrt [3]{b} x+2\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0697781, antiderivative size = 48, 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 \[ \frac{C \log \left (\sqrt [3]{b} x+2\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1613, size = 49, normalized size = 1.02 \[ \frac{C \log{\left (\sqrt [3]{b} x + 2 \right )}}{\sqrt [3]{b}} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\sqrt{3} \left (- \frac{\sqrt [3]{b} x}{3} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*log(b**(1/3)*x + 2)/b**(1/3) - 2*sqrt(3)*C*atan(sqrt(3)*(-b**(1/3)*x/3 + 1/3))
/(3*b**(1/3))

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Mathematica [A]  time = 0.0362854, size = 76, normalized size = 1.58 \[ \frac{C \left (-\log \left (b^{2/3} x^2-2 \sqrt [3]{b} x+4\right )+\log \left (b x^3+8\right )+2 \log \left (\sqrt [3]{b} x+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b} x-1}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 117, normalized size = 2.4 \[{\frac{C\sqrt [3]{8}}{3\,b}\ln \left ( x+\sqrt [3]{8}\sqrt [3]{{b}^{-1}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\sqrt [3]{8}}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{8}\sqrt [3]{{b}^{-1}}+{8}^{{\frac{2}{3}}} \left ({b}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\sqrt [3]{8}\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{8}^{{\frac{2}{3}}}x}{4}{\frac{1}{\sqrt [3]{{b}^{-1}}}}}-1 \right ) } \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}+8 \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*C/b*8^(1/3)/(1/b)^(2/3)*ln(x+8^(1/3)*(1/b)^(1/3))-1/6*C/b*8^(1/3)/(1/b)^(2/3
)*ln(x^2-x*8^(1/3)*(1/b)^(1/3)+8^(2/3)*(1/b)^(2/3))+1/3*C/b*8^(1/3)/(1/b)^(2/3)*
3^(1/2)*arctan(1/3*3^(1/2)*(1/4*8^(2/3)/(1/b)^(1/3)*x-1))+1/3*C/b^(1/3)*ln(b*x^3
+8)

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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((C*b^(2/3)*x^2 + 8*C)/(b*x^3 + 8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247513, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{b x^{2} + 6 \, \sqrt{\frac{1}{3}}{\left (b x - b^{\frac{2}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - 2 \, b^{\frac{2}{3}} x - 2 \, b^{\frac{1}{3}}}{b x^{2} - 2 \, b^{\frac{2}{3}} x + 4 \, b^{\frac{1}{3}}}\right ) + C b^{\frac{2}{3}} \log \left (b x + 2 \, b^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b^{\frac{2}{3}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (b x - b^{\frac{2}{3}}\right )}}{b^{\frac{2}{3}}}\right ) - C b^{\frac{2}{3}} \log \left (b x + 2 \, b^{\frac{2}{3}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(1/3)*C*b*sqrt(-1/b^(2/3))*log((b*x^2 + 6*sqrt(1/3)*(b*x - b^(2/3))*sqrt(-
1/b^(2/3)) - 2*b^(2/3)*x - 2*b^(1/3))/(b*x^2 - 2*b^(2/3)*x + 4*b^(1/3))) + C*b^(
2/3)*log(b*x + 2*b^(2/3)))/b, -(2*sqrt(1/3)*C*b^(2/3)*arctan(-sqrt(1/3)*(b*x - b
^(2/3))/b^(2/3)) - C*b^(2/3)*log(b*x + 2*b^(2/3)))/b]

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Sympy [A]  time = 0.880901, size = 58, normalized size = 1.21 \[ \operatorname{RootSum}{\left (3 t^{3} b^{\frac{5}{3}} - 3 t^{2} C b^{\frac{4}{3}} + t C^{2} b - C^{3} b^{\frac{2}{3}}, \left ( t \mapsto t \log{\left (x + \frac{3 t \sqrt [3]{b} - C}{C \sqrt [3]{b}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(3*_t**3*b**(5/3) - 3*_t**2*C*b**(4/3) + _t*C**2*b - C**3*b**(2/3), Lambd
a(_t, _t*log(x + (3*_t*b**(1/3) - C)/(C*b**(1/3)))))

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GIAC/XCAS [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((C*b^(2/3)*x^2 + 8*C)/(b*x^3 + 8),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError