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

Optimal. Leaf size=53 \[ \frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

[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 = 0.12987, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 C \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

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 = 13.0925, size = 49, 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 = 0.105479, size = 150, normalized size = 2.83 \[ \frac{C \left (b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} \log \left (a-b x^3\right )-2 b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )}{3 a^{2/3} b} \]

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)*Lo
g[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 = 0.009, size = 135, normalized size = 2.6 \[ -{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \]

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)*
arctan(1/3*(1+2/(a/b)^(1/3)*x)*3^(1/2))-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 = 0.233563, size = 81, normalized size = 1.53 \[ -\frac{\sqrt{3}{\left (\sqrt{3} C \log \left (b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - a\right ) - 2 \, C \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right )\right )}}{3 \, b} \]

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 = 0.989133, size = 110, normalized size = 2.08 \[ - \frac{C \left (\log{\left (- \frac{a}{b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} + \frac{\sqrt{3} i \log{\left (\frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} - \frac{\sqrt{3} i \log{\left (\frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \]

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 = 0.24064, size = 204, normalized size = 3.85 \[ \frac{\sqrt{3}{\left (\sqrt{3} a b^{2} i - a b^{2}\right )} C \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{3}} - \frac{{\left (C b^{2} \left (\frac{a}{b}\right )^{\frac{2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} C\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} - \frac{{\left (\sqrt{3} a b^{2} i + 3 \, a b^{2}\right )} C{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{3}} \]

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)*l
n(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)