3.2407 \(\int (a+\frac{b}{\sqrt [3]{x}})^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]

[Out]

((a + b/x^(1/3))^3*x)/a

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Rubi [A]  time = 0.0069336, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 37} \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^2,x]

[Out]

((a + b/x^(1/3))^3*x)/a

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^4} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x}{a}\\ \end{align*}

Mathematica [A]  time = 0.0099199, size = 25, normalized size = 1.56 \[ a^2 x+3 a b x^{2/3}+3 b^2 \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^2,x]

[Out]

3*b^2*x^(1/3) + 3*a*b*x^(2/3) + a^2*x

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( b+a\sqrt [3]{x} \right ) ^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.990099, size = 28, normalized size = 1.75 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)

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Fricas [A]  time = 1.53712, size = 53, normalized size = 3.31 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)

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Sympy [A]  time = 0.15988, size = 24, normalized size = 1.5 \begin{align*} a^{2} x + 3 a b x^{\frac{2}{3}} + 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)

[Out]

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

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Giac [A]  time = 1.20793, size = 28, normalized size = 1.75 \begin{align*} a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)