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

Optimal. Leaf size=50 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]

[Out]

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

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Rubi [A]  time = 0.11925, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5506, size = 46, normalized size = 0.92 \[ \frac{C \log{\left (x + \sqrt [3]{\frac{a}{b}} \right )}}{b} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 x}{3 \sqrt [3]{\frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.10607, size = 146, normalized size = 2.92 \[ \frac{C \left (-b^{2/3} \left (\frac{a}{b}\right )^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )+2 b^{2/3} \left (\frac{a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \left (\frac{a}{b}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 87, normalized size = 1.7 \[{\frac{2\,C}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{C}{3\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) }+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*C*ln(x+(a/b)^(1/3))/b-1/3*C/b*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/3*C/b*3^(1
/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*C/b*ln(b*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((C*x^2 + 2*C*(a/b)^(2/3))/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233033, size = 77, normalized size = 1.54 \[ \frac{\sqrt{3}{\left (\sqrt{3} C \log \left (b x \left (\frac{a}{b}\right )^{\frac{2}{3}} + a\right ) + 2 \, C \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right )\right )}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.94988, size = 100, normalized size = 2. \[ \frac{C \left (\log{\left (\frac{a}{b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} - \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (\frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241073, size = 211, normalized size = 4.22 \[ \frac{\sqrt{3}{\left (\sqrt{3} a b^{2} i + a b^{2}\right )} C \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} - \frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} - \frac{{\left (\sqrt{3} a b^{2} i - 3 \, a b^{2}\right )} C{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*(sqrt(3)*a*b^2*i + a*b^2)*C*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/
(-a/b)^(1/3))/(a*b^3) - 1/3*(C*b^2*(-a/b)^(2/3) + 2*(a*b^2)^(2/3)*C)*(-a/b)^(1/3
)*ln(abs(x - (-a/b)^(1/3)))/(a*b^2) - 1/6*(sqrt(3)*a*b^2*i - 3*a*b^2)*C*ln(x^2 +
x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3)