\(\int \frac {x}{\sqrt [3]{a+b x}} \, dx\) [394]

Optimal result

Integrand size = 11, antiderivative size = 34 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=-\frac {3 a (a+b x)^{2/3}}{2 b^2}+\frac {3 (a+b x)^{5/3}}{5 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=\frac {3 (a+b x)^{5/3}}{5 b^2}-\frac {3 a (a+b x)^{2/3}}{2 b^2} \]

[In]

Int[x/(a + b*x)^(1/3),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=\frac {3 (a+b x)^{2/3} (-3 a+2 b x)}{10 b^2} \]

[In]

Integrate[x/(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(2/3)*(-3*a + 2*b*x))/(10*b^2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
trager	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
risch	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
pseudoelliptic	\(-\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-2 b x +3 a \right )}{10 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {5}{3}}}{5}-\frac {3 a \left (b x +a \right )^{\frac {2}{3}}}{2}}{b^{2}}\)	\(26\)

[In]

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

[Out]

-3/10*(b*x+a)^(2/3)*(-2*b*x+3*a)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=\frac {3 \, {\left (2 \, b x - 3 \, a\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{10 \, b^{2}} \]

[In]

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

[Out]

3/10*(2*b*x - 3*a)*(b*x + a)^(2/3)/b^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (31) = 62\).

Time = 0.76 (sec) , antiderivative size = 162, normalized size of antiderivative = 4.76 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=- \frac {9 a^{\frac {11}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {11}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} - \frac {3 a^{\frac {8}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {9 a^{\frac {8}{3}} b x}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac {6 a^{\frac {5}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} \]

[In]

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

[Out]

-9*a**(11/3)*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(11/3)/(10*a**2*b**2 + 10*a*b**3*x) - 3*a*
*(8/3)*b*x*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x) + 9*a**(8/3)*b*x/(10*a**2*b**2 + 10*a*b**3*x) + 6*a
**(5/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(10*a**2*b**2 + 10*a*b**3*x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}}}{5 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=\frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {5}{3}} - 5 \, {\left (b x + a\right )}^{\frac {2}{3}} a\right )}}{10 \, b^{2}} \]

[In]

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

[Out]

3/10*(2*(b*x + a)^(5/3) - 5*(b*x + a)^(2/3)*a)/b^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {x}{\sqrt [3]{a+b x}} \, dx=-\frac {15\,a\,{\left (a+b\,x\right )}^{2/3}-6\,{\left (a+b\,x\right )}^{5/3}}{10\,b^2} \]

[In]

int(x/(a + b*x)^(1/3),x)

[Out]

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