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

Optimal. Leaf size=57 \[ \frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{-b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{-b}}-\frac{C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]

[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 = 0.112951, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 C \tan ^{-1}\left (\frac{1-\sqrt [3]{-b} x}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{-b}}-\frac{C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]

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 = 13.266, size = 56, normalized size = 0.98 \[ - \frac{C \log{\left (x \sqrt [3]{- b} + 2 \right )}}{\sqrt [3]{- b}} + \frac{2 \sqrt{3} C \operatorname{atan}{\left (\sqrt{3} \left (- \frac{x \sqrt [3]{- b}}{3} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{- b}} \]

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

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Mathematica [A]  time = 0.049111, size = 99, normalized size = 1.74 \[ \frac{C \left (-b^{2/3} \log \left (b^{2/3} x^2+2 \sqrt [3]{b} x+4\right )+2 b^{2/3} \log \left (2-\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b} x+1}{\sqrt{3}}\right )+(-b)^{2/3} \log \left (8-b x^3\right )\right )}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 122, normalized size = 2.1 \[{\frac{C\sqrt [3]{8}}{3\,b}\ln \left ( x-\sqrt [3]{8}\sqrt [3]{{b}^{-1}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\sqrt [3]{8}}{6\,b}\ln \left ({x}^{2}+x\sqrt [3]{8}\sqrt [3]{{b}^{-1}}+{8}^{{\frac{2}{3}}} \left ({b}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\sqrt [3]{8}\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{{8}^{{\frac{2}{3}}}x}{4}{\frac{1}{\sqrt [3]{{b}^{-1}}}}}+1 \right ) } \right ) \left ({b}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}-8 \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \]

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)^(2/3)/b*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 = 0.244853, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{b x^{2} - 6 \, \sqrt{\frac{1}{3}}{\left (b x + \left (-b\right )^{\frac{2}{3}}\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 2 \, \left (-b\right )^{\frac{2}{3}} x + 2 \, \left (-b\right )^{\frac{1}{3}}}{b x^{2} + 2 \, \left (-b\right )^{\frac{2}{3}} x - 4 \, \left (-b\right )^{\frac{1}{3}}}\right ) + C \left (-b\right )^{\frac{2}{3}} \log \left (b x - 2 \, \left (-b\right )^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (b x + \left (-b\right )^{\frac{2}{3}}\right )}}{b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}\right ) - C \left (-b\right )^{\frac{2}{3}} \log \left (b x - 2 \, \left (-b\right )^{\frac{2}{3}}\right )}{b}\right ] \]

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((-b)^(1/3)/b)*log((b*x^2 - 6*sqrt(1/3)*(b*x + (-b)^(2/3))*s
qrt((-b)^(1/3)/b) + 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*sqrt(-(
-b)^(1/3)/b)*arctan(sqrt(1/3)*(b*x + (-b)^(2/3))/(b*sqrt(-(-b)^(1/3)/b))) - C*(-
b)^(2/3)*log(b*x - 2*(-b)^(2/3)))/b]

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Sympy [A]  time = 0.976785, size = 58, normalized size = 1.02 \[ \operatorname{RootSum}{\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac{2}{3}} + t C^{2} \left (- b\right )^{\frac{4}{3}} - C^{3} b, \left ( t \mapsto t \log{\left (- \frac{3 t}{C} + x + \frac{\left (- b\right )^{\frac{2}{3}}}{b} \right )} \right )\right )} \]

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**2 - 3*_t**2*C*b*(-b)**(2/3) + _t*C**2*(-b)**(4/3) - C**3*b, L
ambda(_t, _t*log(-3*_t/C + x + (-b)**(2/3)/b)))

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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