\(\int x \sqrt [3]{a+b x} \, dx\) [373]

Optimal result

Integrand size = 11, antiderivative size = 34 \[ \int x \sqrt [3]{a+b x} \, dx=-\frac {3 a (a+b x)^{4/3}}{4 b^2}+\frac {3 (a+b x)^{7/3}}{7 b^2} \]

[Out]

-3/4*a*(b*x+a)^(4/3)/b^2+3/7*(b*x+a)^(7/3)/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} \[ \int x \sqrt [3]{a+b x} \, dx=\frac {3 (a+b x)^{7/3}}{7 b^2}-\frac {3 a (a+b x)^{4/3}}{4 b^2} \]

[In]

Int[x*(a + b*x)^(1/3),x]

[Out]

(-3*a*(a + b*x)^(4/3))/(4*b^2) + (3*(a + b*x)^(7/3))/(7*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 \sqrt [3]{a+b x}}{b}+\frac {(a+b x)^{4/3}}{b}\right ) \, dx \\ & = -\frac {3 a (a+b x)^{4/3}}{4 b^2}+\frac {3 (a+b x)^{7/3}}{7 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int x \sqrt [3]{a+b x} \, dx=\frac {3 \sqrt [3]{a+b x} \left (-3 a^2+a b x+4 b^2 x^2\right )}{28 b^2} \]

[In]

Integrate[x*(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(1/3)*(-3*a^2 + a*b*x + 4*b^2*x^2))/(28*b^2)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (-4 b x +3 a \right )}{28 b^{2}}\)	\(21\)
pseudoelliptic	\(-\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (-4 b x +3 a \right )}{28 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{2}}\)	\(26\)
trager	\(-\frac {3 \left (-4 b^{2} x^{2}-a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{28 b^{2}}\)	\(32\)
risch	\(-\frac {3 \left (-4 b^{2} x^{2}-a b x +3 a^{2}\right ) \left (b x +a \right )^{\frac {1}{3}}}{28 b^{2}}\)	\(32\)

[In]

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

[Out]

-3/28*(b*x+a)^(4/3)*(-4*b*x+3*a)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int x \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (4 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{28 \, b^{2}} \]

[In]

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

[Out]

3/28*(4*b^2*x^2 + a*b*x - 3*a^2)*(b*x + a)^(1/3)/b^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (31) = 62\).

Time = 1.15 (sec) , antiderivative size = 202, normalized size of antiderivative = 5.94 \[ \int x \sqrt [3]{a+b x} \, dx=- \frac {9 a^{\frac {13}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {9 a^{\frac {13}{3}}}{28 a^{2} b^{2} + 28 a b^{3} x} - \frac {6 a^{\frac {10}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {9 a^{\frac {10}{3}} b x}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {15 a^{\frac {7}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} + \frac {12 a^{\frac {4}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{28 a^{2} b^{2} + 28 a b^{3} x} \]

[In]

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

[Out]

-9*a**(13/3)*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 9*a**(13/3)/(28*a**2*b**2 + 28*a*b**3*x) - 6*a*
*(10/3)*b*x*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 9*a**(10/3)*b*x/(28*a**2*b**2 + 28*a*b**3*x) + 1
5*a**(7/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(28*a**2*b**2 + 28*a*b**3*x) + 12*a**(4/3)*b**3*x**3*(1 + b*x/a)**(1/3
)/(28*a**2*b**2 + 28*a*b**3*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int x \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}}}{7 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a}{4 \, b^{2}} \]

[In]

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

[Out]

3/7*(b*x + a)^(7/3)/b^2 - 3/4*(b*x + a)^(4/3)*a/b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int x \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (\frac {7 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a}{b} + \frac {2 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )}}{b}\right )}}{28 \, b} \]

[In]

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

[Out]

3/28*(7*((b*x + a)^(4/3) - 4*(b*x + a)^(1/3)*a)*a/b + 2*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 14*(b*x + a
)^(1/3)*a^2)/b)/b

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int x \sqrt [3]{a+b x} \, dx=-\frac {21\,a\,{\left (a+b\,x\right )}^{4/3}-12\,{\left (a+b\,x\right )}^{7/3}}{28\,b^2} \]

[In]

int(x*(a + b*x)^(1/3),x)

[Out]

-(21*a*(a + b*x)^(4/3) - 12*(a + b*x)^(7/3))/(28*b^2)