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

Optimal. Leaf size=61 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

[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 = 0.0833869, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

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 = 13.1518, size = 63, normalized size = 1.03 \[ \frac{C \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{\sqrt [3]{b}} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{b}} \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0416317, size = 95, normalized size = 1.56 \[ \frac{C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{b}} \]

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

_______________________________________________________________________________________

Maple [B]  time = 0.005, size = 117, normalized size = 1.9 \[{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]

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^(1/3)*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 = 0.244809, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} b^{\frac{2}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - a b^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} b^{\frac{2}{3}} x + a b^{\frac{1}{3}}}\right ) + C b^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a b^{\frac{2}{3}}\right )}{b}, -\frac{2 \, \sqrt{\frac{1}{3}} C b^{\frac{2}{3}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )}}{a b^{\frac{2}{3}}}\right ) - C b^{\frac{2}{3}} \log \left (a^{\frac{2}{3}} b x + a b^{\frac{2}{3}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.964292, size = 70, normalized size = 1.15 \[ \operatorname{RootSum}{\left (3 t^{3} b^{\frac{5}{3}} - 3 t^{2} C b^{\frac{4}{3}} + t C^{2} b - C^{3} b^{\frac{2}{3}}, \left ( t \mapsto t \log{\left (x + \frac{3 t \sqrt [3]{a} \sqrt [3]{b} - C \sqrt [3]{a}}{2 C \sqrt [3]{b}} \right )} \right )\right )} \]

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**(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*a**(1/3)*b**(1/3) - C*a**(1/3))/(2*C*b**(1/3)))))

_______________________________________________________________________________________

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