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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 21, normalized size = 1.31 \[ a^{2} x + 3 \, a b x^{\frac {2}{3}} + 3 \, b^{2} x^{\frac {1}{3}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 21, normalized size = 1.31 \[ a^{2} x + 3 \, a b x^{\frac {2}{3}} + 3 \, b^{2} x^{\frac {1}{3}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.88 \[ \frac {\left (a \,x^{\frac {1}{3}}+b \right )^{3}}{a} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 21, normalized size = 1.31 \[ a^{2} x + 3 \, a b x^{\frac {2}{3}} + 3 \, b^{2} x^{\frac {1}{3}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 21, normalized size = 1.31 \[ a^2\,x+3\,b^2\,x^{1/3}+3\,a\,b\,x^{2/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^(1/3))^2,x)

[Out]

a^2*x + 3*b^2*x^(1/3) + 3*a*b*x^(2/3)

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 24, normalized size = 1.50 \[ a^{2} x + 3 a b x^{\frac {2}{3}} + 3 b^{2} \sqrt [3]{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________