3.2352 \(\int \frac{1}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=42 \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0533012, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^(-1),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{2} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{3}} + \frac{3 \int ^{\sqrt [3]{x}} x\, dx}{b} - \frac{3 \int ^{\sqrt [3]{x}} a\, dx}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*log(a + b*x**(1/3))/b**3 + 3*Integral(x, (x, x**(1/3)))/b - 3*Integral(a,
(x, x**(1/3)))/b**2

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Mathematica [A]  time = 0.0195126, size = 42, normalized size = 1. \[ \frac{3 a^2 \log \left (a+b \sqrt [3]{x}\right )}{b^3}-\frac{3 a \sqrt [3]{x}}{b^2}+\frac{3 x^{2/3}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 79, normalized size = 1.9 \[{\frac{{a}^{2}\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}}+{\frac{3}{2\,b}{x}^{{\frac{2}{3}}}}+2\,{\frac{{a}^{2}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3}}\ln \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) }-3\,{\frac{a\sqrt [3]{x}}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3)),x)

[Out]

a^2*ln(b^3*x+a^3)/b^3+3/2*x^(2/3)/b+2*a^2*ln(a+b*x^(1/3))/b^3-1/b^3*a^2*ln(b^2*x
^(2/3)-a*b*x^(1/3)+a^2)-3*a*x^(1/3)/b^2

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Maxima [A]  time = 1.4431, size = 59, normalized size = 1.4 \[ \frac{3 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2}}{2 \, b^{3}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )} a}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a^2*log(b*x^(1/3) + a)/b^3 + 3/2*(b*x^(1/3) + a)^2/b^3 - 6*(b*x^(1/3) + a)*a/b
^3

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Fricas [A]  time = 0.215, size = 45, normalized size = 1.07 \[ \frac{3 \,{\left (2 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right ) + b^{2} x^{\frac{2}{3}} - 2 \, a b x^{\frac{1}{3}}\right )}}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.463942, size = 42, normalized size = 1. \[ \begin{cases} \frac{3 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{b^{3}} - \frac{3 a \sqrt [3]{x}}{b^{2}} + \frac{3 x^{\frac{2}{3}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**(1/3)),x)

[Out]

Piecewise((3*a**2*log(a/b + x**(1/3))/b**3 - 3*a*x**(1/3)/b**2 + 3*x**(2/3)/(2*b
), Ne(b, 0)), (x/a, True))

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GIAC/XCAS [A]  time = 0.214726, size = 47, normalized size = 1.12 \[ \frac{3 \, a^{2}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{2}{3}} - 2 \, a x^{\frac{1}{3}}\right )}}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a^2*ln(abs(b*x^(1/3) + a))/b^3 + 3/2*(b*x^(2/3) - 2*a*x^(1/3))/b^2