\(\int x^2 \sqrt [3]{a+b x} \, dx\) [372]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 a^2 (a+b x)^{4/3}}{4 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3} \]

[Out]

3/4*a^2*(b*x+a)^(4/3)/b^3-6/7*a*(b*x+a)^(7/3)/b^3+3/10*(b*x+a)^(10/3)/b^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, 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 = {45} \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 a^2 (a+b x)^{4/3}}{4 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3} \]

[In]

Int[x^2*(a + b*x)^(1/3),x]

[Out]

(3*a^2*(a + b*x)^(4/3))/(4*b^3) - (6*a*(a + b*x)^(7/3))/(7*b^3) + (3*(a + b*x)^(10/3))/(10*b^3)

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^2 \sqrt [3]{a+b x}}{b^2}-\frac {2 a (a+b x)^{4/3}}{b^2}+\frac {(a+b x)^{7/3}}{b^2}\right ) \, dx \\ & = \frac {3 a^2 (a+b x)^{4/3}}{4 b^3}-\frac {6 a (a+b x)^{7/3}}{7 b^3}+\frac {3 (a+b x)^{10/3}}{10 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 (a+b x)^{4/3} \left (9 a^2-12 a b x+14 b^2 x^2\right )}{140 b^3} \]

[In]

Integrate[x^2*(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(4/3)*(9*a^2 - 12*a*b*x + 14*b^2*x^2))/(140*b^3)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (14 b^{2} x^{2}-12 a b x +9 a^{2}\right )}{140 b^{3}}\)	\(32\)
pseudoelliptic	\(\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (14 b^{2} x^{2}-12 a b x +9 a^{2}\right )}{140 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {6 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {3 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {6 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {3 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{3}}\)	\(38\)
trager	\(\frac {3 \left (14 b^{3} x^{3}+2 a \,b^{2} x^{2}-3 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{140 b^{3}}\)	\(43\)
risch	\(\frac {3 \left (14 b^{3} x^{3}+2 a \,b^{2} x^{2}-3 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{140 b^{3}}\)	\(43\)

[In]

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

[Out]

3/140*(b*x+a)^(4/3)*(14*b^2*x^2-12*a*b*x+9*a^2)/b^3

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (14 \, b^{3} x^{3} + 2 \, a b^{2} x^{2} - 3 \, a^{2} b x + 9 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{140 \, b^{3}} \]

[In]

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

[Out]

3/140*(14*b^3*x^3 + 2*a*b^2*x^2 - 3*a^2*b*x + 9*a^3)*(b*x + a)^(1/3)/b^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (49) = 98\).

Time = 1.33 (sec) , antiderivative size = 666, normalized size of antiderivative = 12.57 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {27 a^{\frac {34}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {34}{3}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {31}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {31}{3}} b x}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {28}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {28}{3}} b^{2} x^{2}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {25}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {25}{3}} b^{3} x^{3}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {135 a^{\frac {22}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {132 a^{\frac {19}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} + \frac {42 a^{\frac {16}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{140 a^{8} b^{3} + 420 a^{7} b^{4} x + 420 a^{6} b^{5} x^{2} + 140 a^{5} b^{6} x^{3}} \]

[In]

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

[Out]

27*a**(34/3)*(1 + b*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) -
27*a**(34/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 72*a**(31/3)*b*x*(1
+ b*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) - 81*a**(31/3)*b*
x/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 60*a**(28/3)*b**2*x**2*(1 + b*
x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) - 81*a**(28/3)*b**2*x*
*2/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 60*a**(25/3)*b**3*x**3*(1 + b
*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) - 27*a**(25/3)*b**3*x
**3/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 135*a**(22/3)*b**4*x**4*(1 +
b*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 132*a**(19/3)*b**
5*x**5*(1 + b*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3) + 42*a**
(16/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(140*a**8*b**3 + 420*a**7*b**4*x + 420*a**6*b**5*x**2 + 140*a**5*b**6*x**3
)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{3}} - \frac {6 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2}}{4 \, b^{3}} \]

[In]

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

[Out]

3/10*(b*x + a)^(10/3)/b^3 - 6/7*(b*x + a)^(7/3)*a/b^3 + 3/4*(b*x + a)^(4/3)*a^2/b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {3 \, {\left (\frac {10 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )} a}{b^{2}} + \frac {14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}}{b^{2}}\right )}}{140 \, b} \]

[In]

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

[Out]

3/140*(10*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 14*(b*x + a)^(1/3)*a^2)*a/b^2 + (14*(b*x + a)^(10/3) - 60
*(b*x + a)^(7/3)*a + 105*(b*x + a)^(4/3)*a^2 - 140*(b*x + a)^(1/3)*a^3)/b^2)/b

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int x^2 \sqrt [3]{a+b x} \, dx=\frac {42\,{\left (a+b\,x\right )}^{10/3}-120\,a\,{\left (a+b\,x\right )}^{7/3}+105\,a^2\,{\left (a+b\,x\right )}^{4/3}}{140\,b^3} \]

[In]

int(x^2*(a + b*x)^(1/3),x)

[Out]

(42*(a + b*x)^(10/3) - 120*a*(a + b*x)^(7/3) + 105*a^2*(a + b*x)^(4/3))/(140*b^3)