3.2307 \(\int (a+b \sqrt [3]{x})^3 \, dx\)

Optimal. Leaf size=42 \[ \frac{9}{4} a^2 b x^{4/3}+a^3 x+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.0219194, 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, Rules used = {190, 43} \[ \frac{9}{4} a^2 b x^{4/3}+a^3 x+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^3,x]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^2 (a+b x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=a^3 x+\frac{9}{4} a^2 b x^{4/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.0150406, size = 42, normalized size = 1. \[ \frac{9}{4} a^2 b x^{4/3}+a^3 x+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^3,x]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 33, normalized size = 0.8 \begin{align*}{a}^{3}x+{\frac{9\,b{a}^{2}}{4}{x}^{{\frac{4}{3}}}}+{\frac{9\,{b}^{2}a}{5}{x}^{{\frac{5}{3}}}}+{\frac{{b}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+9/4*a^2*b*x^(4/3)+9/5*a*b^2*x^(5/3)+1/2*b^3*x^2

________________________________________________________________________________________

Maxima [A]  time = 0.969567, size = 43, normalized size = 1.02 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x

________________________________________________________________________________________

Fricas [A]  time = 1.42633, size = 82, normalized size = 1.95 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x

________________________________________________________________________________________

Sympy [A]  time = 1.69621, size = 39, normalized size = 0.93 \begin{align*} a^{3} x + \frac{9 a^{2} b x^{\frac{4}{3}}}{4} + \frac{9 a b^{2} x^{\frac{5}{3}}}{5} + \frac{b^{3} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 9*a**2*b*x**(4/3)/4 + 9*a*b**2*x**(5/3)/5 + b**3*x**2/2

________________________________________________________________________________________

Giac [A]  time = 1.11134, size = 43, normalized size = 1.02 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x