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

Optimal. Leaf size=54 \[ \frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b} \]

[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 = 0.097917, antiderivative size = 54, 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 \[ \frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b} \]

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 = 12.6133, size = 49, normalized size = 0.91 \[ \frac{C \log{\left (x - \sqrt [3]{- \frac{a}{b}} \right )}}{b} - \frac{2 \sqrt{3} C \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- \frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

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 = 0.0845229, size = 149, normalized size = 2.76 \[ \frac{C \left (-b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )+2 b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} b} \]

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)*L
og[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 [B]  time = 0.006, size = 132, normalized size = 2.4 \[{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

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*(-a/b)^(2/3)/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/3*C*(-a/b)^(2/3)/b/(a/b)^(2
/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/3*C*(-a/b)^(2/3)/b/(a/b)^(2/3)*3^(1/2)*a
rctan(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 = 0.236429, size = 80, normalized size = 1.48 \[ \frac{\sqrt{3}{\left (\sqrt{3} C \log \left (b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} + a\right ) + 2 \, C \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right )\right )}}{3 \, b} \]

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 = 0.985062, size = 109, normalized size = 2.02 \[ \frac{C \left (\log{\left (\frac{a}{b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} - \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \]

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 = 0.217677, size = 123, normalized size = 2.28 \[ -\frac{2 \, \sqrt{3} C \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b} - \frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} \]

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]

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