\(\int (a+b \sqrt [3]{x}) x^2 \, dx\) [2287]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {a x^3}{3}+\frac {3}{10} b x^{10/3} \]

[Out]

1/3*a*x^3+3/10*b*x^(10/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {a x^3}{3}+\frac {3}{10} b x^{10/3} \]

[In]

Int[(a + b*x^(1/3))*x^2,x]

[Out]

(a*x^3)/3 + (3*b*x^(10/3))/10

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^2+b x^{7/3}\right ) \, dx \\ & = \frac {a x^3}{3}+\frac {3}{10} b x^{10/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {1}{30} \left (10 a+9 b \sqrt [3]{x}\right ) x^3 \]

[In]

Integrate[(a + b*x^(1/3))*x^2,x]

[Out]

((10*a + 9*b*x^(1/3))*x^3)/30

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {a \,x^{3}}{3}+\frac {3 b \,x^{\frac {10}{3}}}{10}\)	\(14\)
default	\(\frac {a \,x^{3}}{3}+\frac {3 b \,x^{\frac {10}{3}}}{10}\)	\(14\)
trager	\(\frac {a \left (x^{2}+x +1\right ) \left (-1+x \right )}{3}+\frac {3 b \,x^{\frac {10}{3}}}{10}\)	\(20\)

[In]

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

[Out]

1/3*a*x^3+3/10*b*x^(10/3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {3}{10} \, b x^{\frac {10}{3}} + \frac {1}{3} \, a x^{3} \]

[In]

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

[Out]

3/10*b*x^(10/3) + 1/3*a*x^3

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {a x^{3}}{3} + \frac {3 b x^{\frac {10}{3}}}{10} \]

[In]

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

[Out]

a*x**3/3 + 3*b*x**(10/3)/10

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (13) = 26\).

Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 7.84 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10}}{10 \, b^{9}} - \frac {8 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a}{3 \, b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{2}}{2 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{3}}{b^{9}} + \frac {35 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{4}}{b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{5}}{5 \, b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{6}}{b^{9}} - \frac {8 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{7}}{b^{9}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{8}}{2 \, b^{9}} \]

[In]

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

[Out]

3/10*(b*x^(1/3) + a)^10/b^9 - 8/3*(b*x^(1/3) + a)^9*a/b^9 + 21/2*(b*x^(1/3) + a)^8*a^2/b^9 - 24*(b*x^(1/3) + a
)^7*a^3/b^9 + 35*(b*x^(1/3) + a)^6*a^4/b^9 - 168/5*(b*x^(1/3) + a)^5*a^5/b^9 + 21*(b*x^(1/3) + a)^4*a^6/b^9 -
8*(b*x^(1/3) + a)^3*a^7/b^9 + 3/2*(b*x^(1/3) + a)^2*a^8/b^9

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {3}{10} \, b x^{\frac {10}{3}} + \frac {1}{3} \, a x^{3} \]

[In]

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

[Out]

3/10*b*x^(10/3) + 1/3*a*x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt [3]{x}\right ) x^2 \, dx=\frac {a\,x^3}{3}+\frac {3\,b\,x^{10/3}}{10} \]

[In]

int(x^2*(a + b*x^(1/3)),x)

[Out]

(a*x^3)/3 + (3*b*x^(10/3))/10