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 = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ -\frac {\left (a+b \sqrt [3]{x}\right )^3}{a x} \]

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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fricas [A]  time = 0.58, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{\frac {2}{3}} + 3 \, a b x^{\frac {1}{3}} + a^{2}}{x} \]

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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giac [A]  time = 0.17, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{\frac {2}{3}} + 3 \, a b x^{\frac {1}{3}} + a^{2}}{x} \]

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

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maple [A]  time = 0.00, size = 25, normalized size = 1.32 \[ -\frac {3 b^{2}}{x^{\frac {1}{3}}}-\frac {3 a b}{x^{\frac {2}{3}}}-\frac {a^{2}}{x} \]

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 = 0.93, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{\frac {2}{3}} + 3 \, a b x^{\frac {1}{3}} + a^{2}}{x} \]

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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mupad [B]  time = 0.03, size = 24, normalized size = 1.26 \[ -\frac {a^2}{x}-\frac {3\,b^2}{x^{1/3}}-\frac {3\,a\,b}{x^{2/3}} \]

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*b^2)/x^(1/3) - (3*a*b)/x^(2/3)

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sympy [A]  time = 0.63, size = 26, normalized size = 1.37 \[ - \frac {a^{2}}{x} - \frac {3 a b}{x^{\frac {2}{3}}} - \frac {3 b^{2}}{\sqrt [3]{x}} \]

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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