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

Optimal. Leaf size=71 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]

[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 = 0.158123, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{C \log \left (\sqrt [3]{\frac{a}{b}}+x\right )}{b}-\frac{2 \left (\frac{a}{b}\right )^{2/3} \left (C \sqrt [3]{\frac{a}{b}}+B\right ) \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} a} \]

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 = 17.3761, size = 54, normalized size = 0.76 \[ \frac{C \log{\left (x + \sqrt [3]{\frac{a}{b}} \right )}}{b} - \frac{2 \sqrt{3} \left (\frac{B}{\sqrt [3]{\frac{a}{b}}} + C\right ) \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 x}{3 \sqrt [3]{\frac{a}{b}}} + \frac{1}{3}\right ) \right )}}{3 b} \]

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 = 0.634627, size = 247, normalized size = 3.48 \[ \frac{\sqrt [3]{b} \left (a^{2/3} B-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )-a^{2/3} B\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} \sqrt [3]{\frac{a}{b}} \left (2 C \sqrt [3]{\frac{a}{b}}+B\right )+\sqrt [3]{a} B\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+2 a C \log \left (a+b x^3\right )}{6 a b} \]

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 [A]  time = 0.007, size = 121, normalized size = 1.7 \[{\frac{2\,C}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{C}{3\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) }+{\frac{2\,C\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) }+{\frac{2\,B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

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*ln(x+(a/b)^(1/3))/b-1/3*C/b*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/3*C/b*3^(1
/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+2/3*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 = 0.248145, size = 351, normalized size = 4.94 \[ -\frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + B b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (a b^{2}\right )^{\frac{1}{3}} B b + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} + \frac{\sqrt{3}{\left ({\left (9 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} - 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B + 18 \,{\left (\sqrt{3} a^{2} b^{3} i + a^{2} b^{3}\right )} C\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{54 \, a^{2} b^{4}} + \frac{{\left ({\left (27 \, \left (-a^{2} b^{4}\right )^{\frac{1}{3}} a b^{2} + 27^{\frac{5}{6}} \left (-a^{2} b^{4}\right )^{\frac{5}{6}}\right )} B - 18 \,{\left (\sqrt{3} a^{2} b^{3} i - 3 \, a^{2} b^{3}\right )} C\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{108 \, a^{2} b^{4}} \]

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*((2
7*(-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)