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

Optimal. Leaf size=78 \[ \frac{2 \left (B-C \sqrt [3]{-\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b} \]

[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

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Rubi [A]  time = 0.175848, antiderivative size = 78, 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 \[ \frac{2 \left (B-C \sqrt [3]{-\frac{a}{b}}\right ) \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b \sqrt [3]{-\frac{a}{b}}}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b} \]

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

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Rubi in Sympy [A]  time = 19.4657, size = 60, normalized size = 0.77 \[ \frac{C \log{\left (x - \sqrt [3]{- \frac{a}{b}} \right )}}{b} - \frac{2 \sqrt{3} \left (- \frac{B}{\sqrt [3]{- \frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- \frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

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 = 0.51506, size = 253, normalized size = 3.24 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (B-2 C \sqrt [3]{-\frac{a}{b}}\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{b} \left (a^{2/3} B+\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (B-2 C \sqrt [3]{-\frac{a}{b}}\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{-\frac{a}{b}} \left (2 C \sqrt [3]{-\frac{a}{b}}-B\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]

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[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 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 [B]  time = 0.007, size = 340, normalized size = 4.4 \[{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \sqrt [3]{-{\frac{a}{b}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

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*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3
/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*(-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*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 + 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 = 0.223189, size = 180, normalized size = 2.31 \[ -\frac{2 \, \sqrt{3}{\left (C a b + \left (-a b^{2}\right )^{\frac{2}{3}} B\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - \left (-a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} \]

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)