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

Optimal. Leaf size=70 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.116095, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.082 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.7928, size = 70, normalized size = 1. \[ \frac{C \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{\sqrt [3]{b}} - \frac{2 \sqrt{3} \left (\frac{B}{\sqrt [3]{a}} + \frac{C}{\sqrt [3]{b}}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0986351, size = 122, normalized size = 1.74 \[ \frac{\sqrt [3]{a} C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )\right )-2 \sqrt{3} \left (\sqrt [3]{a} C+\sqrt [3]{b} B\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 310, normalized size = 4.4 \[{\frac{B}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B/b^(2/3)*a^(1/3)/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+2/3*C*a^(2/3)/b/(a/b)^(2/3)*
ln(x+(a/b)^(1/3))-1/6*B/b^(2/3)*a^(1/3)/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))-1/3*C*a^(2/3)/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*B/b^(2/3
)*a^(1/3)/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+2/3*C*a^(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/b^(1/3)/
(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*B/b^(1/3)/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/
b)^(2/3))+1/3*B/b^(1/3)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-
1))+1/3*C/b^(1/3)*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*b^(2/3)*x^2 + B*b^(2/3)*x + 2*C*a^(2/3) + B*a^(1/3)*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*b^(2/3)*x^2 + B*b^(2/3)*x + 2*C*a^(2/3) + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [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*b^(2/3)*x^2 + B*b^(2/3)*x + 2*C*a^(2/3) + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="giac")

[Out]

Timed out