\(\int (a+\frac {b}{\sqrt [3]{x}}) x^3 \, dx\) [2395]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {3}{11} b x^{11/3}+\frac {a x^4}{4} \]

[Out]

3/11*b*x^(11/3)+1/4*a*x^4

Rubi [A] (verified)

Time = 0.01 (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} \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {a x^4}{4}+\frac {3}{11} b x^{11/3} \]

[In]

Int[(a + b/x^(1/3))*x^3,x]

[Out]

(3*b*x^(11/3))/11 + (a*x^4)/4

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 (b x^{8/3}+a x^3\right ) \, dx \\ & = \frac {3}{11} b x^{11/3}+\frac {a x^4}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {3}{11} b x^{11/3}+\frac {a x^4}{4} \]

[In]

Integrate[(a + b/x^(1/3))*x^3,x]

[Out]

(3*b*x^(11/3))/11 + (a*x^4)/4

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {3 b \,x^{\frac {11}{3}}}{11}+\frac {a \,x^{4}}{4}\)	\(14\)
default	\(\frac {3 b \,x^{\frac {11}{3}}}{11}+\frac {a \,x^{4}}{4}\)	\(14\)
trager	\(\frac {a \left (x^{3}+x^{2}+x +1\right ) \left (-1+x \right )}{4}+\frac {3 b \,x^{\frac {11}{3}}}{11}\)	\(23\)

[In]

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

[Out]

3/11*b*x^(11/3)+1/4*a*x^4

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {1}{4} \, a x^{4} + \frac {3}{11} \, b x^{\frac {11}{3}} \]

[In]

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

[Out]

1/4*a*x^4 + 3/11*b*x^(11/3)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {a x^{4}}{4} + \frac {3 b x^{\frac {11}{3}}}{11} \]

[In]

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

[Out]

a*x**4/4 + 3*b*x**(11/3)/11

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {1}{44} \, {\left (11 \, a + \frac {12 \, b}{x^{\frac {1}{3}}}\right )} x^{4} \]

[In]

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

[Out]

1/44*(11*a + 12*b/x^(1/3))*x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {1}{4} \, a x^{4} + \frac {3}{11} \, b x^{\frac {11}{3}} \]

[In]

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

[Out]

1/4*a*x^4 + 3/11*b*x^(11/3)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^3 \, dx=\frac {a\,x^4}{4}+\frac {3\,b\,x^{11/3}}{11} \]

[In]

int(x^3*(a + b/x^(1/3)),x)

[Out]

(a*x^4)/4 + (3*b*x^(11/3))/11