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

Optimal. Leaf size=76 \[ \frac{2 \left (C \sqrt [3]{-\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.169493, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089 \[ \frac{2 \left (C \sqrt [3]{-\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}-\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}+x\right )}{b} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.5922, size = 60, normalized size = 0.79 \[ - \frac{C \log{\left (x + \sqrt [3]{- \frac{a}{b}} \right )}}{b} + \frac{2 \sqrt{3} \left (\frac{B}{\sqrt [3]{- \frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 x}{3 \sqrt [3]{- \frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

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)

_______________________________________________________________________________________

Mathematica [B]  time = 0.446999, size = 288, normalized size = 3.79 \[ -\frac{\left (-a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}-2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{2/3}}-\frac{\left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}+2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a b^{2/3}}-\frac{\left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} B \sqrt [3]{-\frac{a}{b}}-2 \sqrt [3]{a} \sqrt [3]{b} C \left (-\frac{a}{b}\right )^{2/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a b^{2/3}}-\frac{C \log \left (a-b x^3\right )}{3 b} \]

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]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 345, normalized size = 4.5 \[ -{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{C\ln \left ( b{x}^{3}-a \right ) }{3\,b}} \]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.247676, size = 344, normalized size = 4.53 \[ -\frac{{\left (C b^{2} \left (\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (-a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} C\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} + \frac{\sqrt{3}{\left ({\left (9 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} + 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B + 18 \,{\left (\sqrt{3} a^{2} b^{3} i - a^{2} b^{3}\right )} C\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{54 \, a^{2} b^{4}} + \frac{{\left ({\left (27 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} - 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B - 18 \,{\left (\sqrt{3} a^{2} b^{3} i + 3 \, a^{2} b^{3}\right )} C\right )}{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{108 \, a^{2} b^{4}} \]

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]

-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) + 1/54*sqrt(3)*((9*(-a^2*b^4
)^(1/3)*a*b^2 + 27^(5/6)*(-a^2*b^4)^(5/6))*B + 18*(sqrt(3)*a^2*b^3*i - a^2*b^3)*
C)*arctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b^4) + 1/108*((27*(-
a^2*b^4)^(1/3)*a*b^2 - 27^(5/6)*(-a^2*b^4)^(5/6))*B - 18*(sqrt(3)*a^2*b^3*i + 3*
a^2*b^3)*C)*ln(x^2 + x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b^4)