\(\int \frac {a+b \sqrt [3]{x}}{x^2} \, dx\) [2291]

Optimal result

Integrand size = 13, antiderivative size = 17 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {a}{x}-\frac {3 b}{2 x^{2/3}} \]

[Out]

-a/x-3/2*b/x^(2/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, 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} \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {a}{x}-\frac {3 b}{2 x^{2/3}} \]

[In]

Int[(a + b*x^(1/3))/x^2,x]

[Out]

-(a/x) - (3*b)/(2*x^(2/3))

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 (\frac {a}{x^2}+\frac {b}{x^{5/3}}\right ) \, dx \\ & = -\frac {a}{x}-\frac {3 b}{2 x^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=\frac {-2 a-3 b \sqrt [3]{x}}{2 x} \]

[In]

Integrate[(a + b*x^(1/3))/x^2,x]

[Out]

(-2*a - 3*b*x^(1/3))/(2*x)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(-\frac {a}{x}-\frac {3 b}{2 x^{\frac {2}{3}}}\)	\(14\)
default	\(-\frac {a}{x}-\frac {3 b}{2 x^{\frac {2}{3}}}\)	\(14\)
trager	\(\frac {a \left (-1+x \right )}{x}-\frac {3 b}{2 x^{\frac {2}{3}}}\)	\(16\)

[In]

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

[Out]

-a/x-3/2*b/x^(2/3)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

[In]

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

[Out]

-1/2*(3*b*x^(1/3) + 2*a)/x

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=- \frac {a}{x} - \frac {3 b}{2 x^{\frac {2}{3}}} \]

[In]

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

[Out]

-a/x - 3*b/(2*x**(2/3))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

[In]

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

[Out]

-1/2*(3*b*x^(1/3) + 2*a)/x

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {3 \, b x^{\frac {1}{3}} + 2 \, a}{2 \, x} \]

[In]

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

[Out]

-1/2*(3*b*x^(1/3) + 2*a)/x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \sqrt [3]{x}}{x^2} \, dx=-\frac {a}{x}-\frac {3\,b}{2\,x^{2/3}} \]

[In]

int((a + b*x^(1/3))/x^2,x)

[Out]

- a/x - (3*b)/(2*x^(2/3))