3.47 \(\int \frac{-\sqrt [3]{\frac{a}{b}} B+2 \left (\frac{a}{b}\right )^{2/3} C+B x+C x^2}{a-b x^3} \, dx\)

Optimal. Leaf size=75 \[ -\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (B-C \sqrt [3]{\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} a}-\frac{C \log \left (\sqrt [3]{\frac{a}{b}}-x\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.174261, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (B-C \sqrt [3]{\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} a}-\frac{C \log \left (\sqrt [3]{\frac{a}{b}}-x\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[(-((a/b)^(1/3)*B) + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a - b*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.804, size = 54, normalized size = 0.72 \[ - \frac{C \log{\left (x - \sqrt [3]{\frac{a}{b}} \right )}}{b} + \frac{2 \sqrt{3} \left (- \frac{B}{\sqrt [3]{\frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{\frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-(a/b)**(1/3)*B+2*(a/b)**(2/3)*C+B*x+C*x**2)/(-b*x**3+a),x)

[Out]

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

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Mathematica [B]  time = 0.608016, size = 244, normalized size = 3.25 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}-B\right )\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}-B\right )\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (B-2 C \sqrt [3]{\frac{a}{b}}\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-2 a C \log \left (a-b x^3\right )}{6 a b} \]

Antiderivative was successfully verified.

[In]  Integrate[(-((a/b)^(1/3)*B) + 2*(a/b)^(2/3)*C + B*x + C*x^2)/(a - b*x^3),x]

[Out]

(-2*Sqrt[3]*a^(1/3)*b^(1/3)*(a^(1/3)*B + (a/b)^(1/3)*b^(1/3)*(B - 2*(a/b)^(1/3)*
C))*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*b^(1/3)*(a^(2/3)*B + a^(1/3)
*(a/b)^(1/3)*b^(1/3)*(-B + 2*(a/b)^(1/3)*C))*Log[a^(1/3) - b^(1/3)*x] + b^(1/3)*
(a^(2/3)*B + a^(1/3)*(a/b)^(1/3)*b^(1/3)*(-B + 2*(a/b)^(1/3)*C))*Log[a^(2/3) + a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 2*a*C*Log[a - b*x^3])/(6*a*b)

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Maple [A]  time = 0.008, size = 124, normalized size = 1.7 \[ -{\frac{2\,C}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{C}{3\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) }-{\frac{2\,B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-(a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+C*x^2)/(-b*x^3+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [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*x^2 + B*x + 2*C*(a/b)^(2/3) - B*(a/b)^(1/3))/(b*x^3 - a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: PolynomialDivisionFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: PolynomialDivisionFailed

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GIAC/XCAS [A]  time = 0.223387, size = 169, normalized size = 2.25 \[ \frac{2 \, \sqrt{3}{\left (C a b - \left (a b^{2}\right )^{\frac{2}{3}} B\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{{\left (C b^{2} \left (\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (\frac{a}{b}\right )^{\frac{1}{3}} - \left (a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(C*x^2 + B*x + 2*C*(a/b)^(2/3) - B*(a/b)^(1/3))/(b*x^3 - a),x, algorithm="giac")

[Out]

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