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

Optimal. Leaf size=39 \[ -\frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.0609289, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.1477, size = 44, normalized size = 1.13 \[ - \frac{2 \sqrt{3} B \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0368662, size = 35, normalized size = 0.9 \[ -\frac{2 B \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.005, size = 195, normalized size = 5. \[{\frac{B}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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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((B*b^(2/3)*x + B*a^(1/3)*b^(1/3))/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247734, size = 1, normalized size = 0.03 \[ \left [\sqrt{\frac{1}{3}} B \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} b^{\frac{2}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b^{\frac{2}{3}} x - a^{\frac{4}{3}} b^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - a b^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} b^{\frac{2}{3}} x + a b^{\frac{1}{3}}}\right ), -\frac{2 \, \sqrt{\frac{1}{3}} B \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a b^{\frac{2}{3}}\right )}}{a b^{\frac{2}{3}}}\right )}{a^{\frac{1}{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[sqrt(1/3)*B*sqrt(-1/a^(2/3))*log((2*a^(1/3)*b*x^2 - 2*a^(2/3)*b^(2/3)*x + 3*sqr
t(1/3)*(2*a*b^(2/3)*x - a^(4/3)*b^(1/3))*sqrt(-1/a^(2/3)) - a*b^(1/3))/(a^(1/3)*
b*x^2 - a^(2/3)*b^(2/3)*x + a*b^(1/3))), -2*sqrt(1/3)*B*arctan(-sqrt(1/3)*(2*a^(
2/3)*b*x - a*b^(2/3))/(a*b^(2/3)))/a^(1/3)]

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Sympy [A]  time = 0.821074, size = 88, normalized size = 2.26 \[ \frac{B \left (- \frac{\sqrt{3} i \log{\left (x + \frac{- B \sqrt [3]{a} - \sqrt{3} i B \sqrt [3]{a}}{2 B \sqrt [3]{b}} \right )}}{3} + \frac{\sqrt{3} i \log{\left (x + \frac{- B \sqrt [3]{a} + \sqrt{3} i B \sqrt [3]{a}}{2 B \sqrt [3]{b}} \right )}}{3}\right )}{\sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(-sqrt(3)*I*log(x + (-B*a**(1/3) - sqrt(3)*I*B*a**(1/3))/(2*B*b**(1/3)))/3 + s
qrt(3)*I*log(x + (-B*a**(1/3) + sqrt(3)*I*B*a**(1/3))/(2*B*b**(1/3)))/3)/a**(1/3
)

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GIAC/XCAS [A]  time = 0.225475, size = 65, normalized size = 1.67 \[ \frac{2 \, \sqrt{3} B b^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b^{\frac{2}{3}} x - a^{\frac{1}{3}} b^{\frac{1}{3}}\right )}}{3 \, \sqrt{a^{\frac{2}{3}} b^{\frac{2}{3}}}}\right )}{3 \, \sqrt{a^{\frac{2}{3}} b^{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*B*b^(1/3)*arctan(1/3*sqrt(3)*(2*b^(2/3)*x - a^(1/3)*b^(1/3))/sqrt(a^
(2/3)*b^(2/3)))/sqrt(a^(2/3)*b^(2/3))