3.2300 \(\int \frac{(a+b \sqrt [3]{x})^2}{x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

[Out]

-((a + b*x^(1/3))^3/(a*x))

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Rubi [A]  time = 0.003261, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^2/x^2,x]

[Out]

-((a + b*x^(1/3))^3/(a*x))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x^2} \, dx &=-\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x}\\ \end{align*}

Mathematica [A]  time = 0.0037742, size = 19, normalized size = 1. \[ -\frac{\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^2/x^2,x]

[Out]

-((a + b*x^(1/3))^3/(a*x))

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Maple [A]  time = 0.006, size = 25, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{x}}-3\,{\frac{ab}{{x}^{2/3}}}-3\,{\frac{{b}^{2}}{\sqrt [3]{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 0.972983, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44842, size = 57, normalized size = 3. \begin{align*} -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.82251, size = 26, normalized size = 1.37 \begin{align*} - \frac{a^{2}}{x} - \frac{3 a b}{x^{\frac{2}{3}}} - \frac{3 b^{2}}{\sqrt [3]{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**2/x**2,x)

[Out]

-a**2/x - 3*a*b/x**(2/3) - 3*b**2/x**(1/3)

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Giac [A]  time = 1.11324, size = 32, normalized size = 1.68 \begin{align*} -\frac{3 \, b^{2} x^{\frac{2}{3}} + 3 \, a b x^{\frac{1}{3}} + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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