\(\int \frac {a+b x}{\sqrt [3]{x}} \, dx\) [654]

Optimal result

Integrand size = 11, antiderivative size = 21 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{2} a x^{2/3}+\frac {3}{5} b x^{5/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{2} a x^{2/3}+\frac {3}{5} b x^{5/3} \]

[In]

Int[(a + b*x)/x^(1/3),x]

[Out]

(3*a*x^(2/3))/2 + (3*b*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])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{10} x^{2/3} (5 a+2 b x) \]

[In]

Integrate[(a + b*x)/x^(1/3),x]

[Out]

(3*x^(2/3)*(5*a + 2*b*x))/10

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62

method	result	size
trager	\(\left (\frac {3 b x}{5}+\frac {3 a}{2}\right ) x^{\frac {2}{3}}\)	\(13\)
gosper	\(\frac {3 x^{\frac {2}{3}} \left (2 b x +5 a \right )}{10}\)	\(14\)
derivativedivides	\(\frac {3 a \,x^{\frac {2}{3}}}{2}+\frac {3 b \,x^{\frac {5}{3}}}{5}\)	\(14\)
default	\(\frac {3 a \,x^{\frac {2}{3}}}{2}+\frac {3 b \,x^{\frac {5}{3}}}{5}\)	\(14\)
risch	\(\frac {3 x^{\frac {2}{3}} \left (2 b x +5 a \right )}{10}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{10} \, {\left (2 \, b x + 5 \, a\right )} x^{\frac {2}{3}} \]

[In]

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

[Out]

3/10*(2*b*x + 5*a)*x^(2/3)

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3 a x^{\frac {2}{3}}}{2} + \frac {3 b x^{\frac {5}{3}}}{5} \]

[In]

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

[Out]

3*a*x**(2/3)/2 + 3*b*x**(5/3)/5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{5} \, b x^{\frac {5}{3}} + \frac {3}{2} \, a x^{\frac {2}{3}} \]

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3}{5} \, b x^{\frac {5}{3}} + \frac {3}{2} \, a x^{\frac {2}{3}} \]

[In]

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

[Out]

3/5*b*x^(5/3) + 3/2*a*x^(2/3)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x}{\sqrt [3]{x}} \, dx=\frac {3\,x^{2/3}\,\left (5\,a+2\,b\,x\right )}{10} \]

[In]

int((a + b*x)/x^(1/3),x)

[Out]

(3*x^(2/3)*(5*a + 2*b*x))/10