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

Optimal. Leaf size=41 \[ \frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

[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 = 0.101456, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

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 = 13.6531, size = 44, normalized size = 1.07 \[ \frac{2 \sqrt{3} B \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{- b}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a}} \]

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*x*(-b)**(1/3)/3)/a**(1/3))/(3*a**(1/3))

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Mathematica [B]  time = 0.0973164, size = 129, normalized size = 3.15 \[ \frac{\sqrt [3]{-b} B \left (\left (\sqrt [3]{-b}+\sqrt [3]{b}\right ) \left (2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )+2 \sqrt{3} \left (\sqrt [3]{-b}-\sqrt [3]{b}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 228, normalized size = 5.6 \[{\frac{B\sqrt [3]{-1}}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B\sqrt [3]{-1}}{6}\sqrt [3]{a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt [3]{-1}\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt [3]{-1}}{3}\sqrt [3]{-b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B\sqrt [3]{-1}}{6}\sqrt [3]{-b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B\sqrt [3]{-1}\sqrt{3}}{3}\sqrt [3]{-b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

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)*(-1)^(1/3)*a^(1/3)/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6*B/b^(2/3)*(-1
)^(1/3)*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)*(-1)
^(1/3)*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^(2/3)*(-1)^(1/3)*(-b)^(1/3)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6*B/b^(2/3)*(-1)^
(1/3)*(-b)^(1/3)/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/3*B/b^(2/3)*(-1
)^(1/3)*(-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 = 0.242509, size = 1, normalized size = 0.02 \[ \left [\sqrt{\frac{1}{3}} B \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, a^{\frac{1}{3}} b x^{2} - 2 \, a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a \left (-b\right )^{\frac{2}{3}} x + a^{\frac{4}{3}} \left (-b\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} + a \left (-b\right )^{\frac{1}{3}}}{a^{\frac{1}{3}} b x^{2} - a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x - a \left (-b\right )^{\frac{1}{3}}}\right ), -\frac{2 \, \sqrt{\frac{1}{3}} B \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b x - a \left (-b\right )^{\frac{2}{3}}\right )}}{a \left (-b\right )^{\frac{2}{3}}}\right )}{a^{\frac{1}{3}}}\right ] \]

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*
sqrt(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 = 0.928607, size = 105, normalized size = 2.56 \[ - \frac{B \left (- \frac{\sqrt{3} i \log{\left (- \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} - \frac{\sqrt{3} i \sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + \frac{\sqrt{3} i \sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )}}{3}\right )}{\sqrt [3]{a}} \]

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

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GIAC/XCAS [A]  time = 0.225563, size = 78, normalized size = 1.9 \[ \frac{2 \, \sqrt{3} B b \arctan \left (-\frac{\sqrt{3}{\left (2 \, \left (-b\right )^{\frac{2}{3}} x + a^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}}\right )}}{3 \, \sqrt{a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}}}}\right )}{3 \, \sqrt{a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}}} \left (-b\right )^{\frac{2}{3}}} \]

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*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))*(-b)^(2/3))