\(\int (a+b \sqrt [3]{x})^2 \, dx\) [2298]

Optimal result

Integrand size = 11, antiderivative size = 29 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=a^2 x+\frac {3}{2} a b x^{4/3}+\frac {3}{5} b^2 x^{5/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {196, 45} \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=a^2 x+\frac {3}{2} a b x^{4/3}+\frac {3}{5} b^2 x^{5/3} \]

[In]

Int[(a + b*x^(1/3))^2,x]

[Out]

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

Rule 45

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])

Rule 196

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=\frac {1}{10} \left (10 a^2 x+15 a b x^{4/3}+6 b^2 x^{5/3}\right ) \]

[In]

Integrate[(a + b*x^(1/3))^2,x]

[Out]

(10*a^2*x + 15*a*b*x^(4/3) + 6*b^2*x^(5/3))/10

Maple [A] (verified)

Time = 5.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(a^{2} x +\frac {3 a b \,x^{\frac {4}{3}}}{2}+\frac {3 b^{2} x^{\frac {5}{3}}}{5}\)	\(22\)
default	\(a^{2} x +\frac {3 a b \,x^{\frac {4}{3}}}{2}+\frac {3 b^{2} x^{\frac {5}{3}}}{5}\)	\(22\)
trager	\(a^{2} \left (-1+x \right )+\frac {3 a b \,x^{\frac {4}{3}}}{2}+\frac {3 b^{2} x^{\frac {5}{3}}}{5}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=\frac {3}{5} \, b^{2} x^{\frac {5}{3}} + \frac {3}{2} \, a b x^{\frac {4}{3}} + a^{2} x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=a^{2} x + \frac {3 a b x^{\frac {4}{3}}}{2} + \frac {3 b^{2} x^{\frac {5}{3}}}{5} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=\frac {3}{5} \, b^{2} x^{\frac {5}{3}} + \frac {3}{2} \, a b x^{\frac {4}{3}} + a^{2} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=\frac {3}{5} \, b^{2} x^{\frac {5}{3}} + \frac {3}{2} \, a b x^{\frac {4}{3}} + a^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt [3]{x}\right )^2 \, dx=a^2\,x+\frac {3\,b^2\,x^{5/3}}{5}+\frac {3\,a\,b\,x^{4/3}}{2} \]

[In]

int((a + b*x^(1/3))^2,x)

[Out]

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