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

Optimal. Leaf size=70 \[ \frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{-b}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.12748, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{C \log \left (\sqrt [3]{a}-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.8648, size = 70, normalized size = 1. \[ \frac{C \log{\left (\sqrt [3]{a} - x \sqrt [3]{- b} \right )}}{\sqrt [3]{- b}} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{- b}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{- b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.057948, size = 116, normalized size = 1.66 \[ -\frac{C \left (-b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+(-b)^{2/3} \log \left (a+b x^3\right )\right )}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 122, normalized size = 1.7 \[ -{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b} \left ( -b \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*C*a^(2/3)/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+1/3*C*a^(2/3)/b/(a/b)^(2/3)*ln(x^
2-x*(a/b)^(1/3)+(a/b)^(2/3))-2/3*C*a^(2/3)/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1
/2)*(2/(a/b)^(1/3)*x-1))-1/3*C*(-b)^(2/3)/b*ln(b*x^3+a)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.249605, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a \left (-b\right )^{\frac{2}{3}}\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + a \left (-b\right )^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - a \left (-b\right )^{\frac{1}{3}}}\right ) - C \left (-b\right )^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a \left (-b\right )^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a \left (-b\right )^{\frac{2}{3}}\right )}}{a b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}\right ) + C \left (-b\right )^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a \left (-b\right )^{\frac{2}{3}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.06539, size = 73, normalized size = 1.04 \[ - \operatorname{RootSum}{\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac{2}{3}} + t C^{2} \left (- b\right )^{\frac{4}{3}} - C^{3} b, \left ( t \mapsto t \log{\left (\frac{3 t \sqrt [3]{a}}{2 C} - \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*a**(2/3)*C-(-b)**(2/3)*C*x**2)/(b*x**3+a),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,
Lambda(_t, _t*log(3*_t*a**(1/3)/(2*C) - a**(1/3)*(-b)**(2/3)/(2*b) + x)))

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out