\(\int x^2 (a+b x)^{2/3} \, dx\) [379]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 a^2 (a+b x)^{5/3}}{5 b^3}-\frac {3 a (a+b x)^{8/3}}{4 b^3}+\frac {3 (a+b x)^{11/3}}{11 b^3} \]

[Out]

3/5*a^2*(b*x+a)^(5/3)/b^3-3/4*a*(b*x+a)^(8/3)/b^3+3/11*(b*x+a)^(11/3)/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} \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 a^2 (a+b x)^{5/3}}{5 b^3}+\frac {3 (a+b x)^{11/3}}{11 b^3}-\frac {3 a (a+b x)^{8/3}}{4 b^3} \]

[In]

Int[x^2*(a + b*x)^(2/3),x]

[Out]

(3*a^2*(a + b*x)^(5/3))/(5*b^3) - (3*a*(a + b*x)^(8/3))/(4*b^3) + (3*(a + b*x)^(11/3))/(11*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 (a+b x)^{2/3}}{b^2}-\frac {2 a (a+b x)^{5/3}}{b^2}+\frac {(a+b x)^{8/3}}{b^2}\right ) \, dx \\ & = \frac {3 a^2 (a+b x)^{5/3}}{5 b^3}-\frac {3 a (a+b x)^{8/3}}{4 b^3}+\frac {3 (a+b x)^{11/3}}{11 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 (a+b x)^{5/3} \left (9 a^2-15 a b x+20 b^2 x^2\right )}{220 b^3} \]

[In]

Integrate[x^2*(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3)*(9*a^2 - 15*a*b*x + 20*b^2*x^2))/(220*b^3)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (20 b^{2} x^{2}-15 a b x +9 a^{2}\right )}{220 b^{3}}\)	\(32\)
pseudoelliptic	\(\frac {3 \left (b x +a \right )^{\frac {5}{3}} \left (20 b^{2} x^{2}-15 a b x +9 a^{2}\right )}{220 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {11}{3}}}{11}-\frac {3 a \left (b x +a \right )^{\frac {8}{3}}}{4}+\frac {3 a^{2} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {11}{3}}}{11}-\frac {3 a \left (b x +a \right )^{\frac {8}{3}}}{4}+\frac {3 a^{2} \left (b x +a \right )^{\frac {5}{3}}}{5}}{b^{3}}\)	\(38\)
trager	\(\frac {3 \left (20 b^{3} x^{3}+5 a \,b^{2} x^{2}-6 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {2}{3}}}{220 b^{3}}\)	\(43\)
risch	\(\frac {3 \left (20 b^{3} x^{3}+5 a \,b^{2} x^{2}-6 a^{2} b x +9 a^{3}\right ) \left (b x +a \right )^{\frac {2}{3}}}{220 b^{3}}\)	\(43\)

[In]

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

[Out]

3/220*(b*x+a)^(5/3)*(20*b^2*x^2-15*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 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 \, {\left (20 \, b^{3} x^{3} + 5 \, a b^{2} x^{2} - 6 \, a^{2} b x + 9 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{220 \, b^{3}} \]

[In]

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

[Out]

3/220*(20*b^3*x^3 + 5*a*b^2*x^2 - 6*a^2*b*x + 9*a^3)*(b*x + a)^(2/3)/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 = 1.85 (sec) , antiderivative size = 666, normalized size of antiderivative = 12.57 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {27 a^{\frac {35}{3}} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {35}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {63 a^{\frac {32}{3}} b x \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {32}{3}} b x}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {42 a^{\frac {29}{3}} b^{2} x^{2} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {29}{3}} b^{2} x^{2}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {78 a^{\frac {26}{3}} b^{3} x^{3} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {26}{3}} b^{3} x^{3}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {207 a^{\frac {23}{3}} b^{4} x^{4} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {195 a^{\frac {20}{3}} b^{5} x^{5} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {17}{3}} b^{6} x^{6} \left (1 + \frac {b x}{a}\right )^{\frac {2}{3}}}{220 a^{8} b^{3} + 660 a^{7} b^{4} x + 660 a^{6} b^{5} x^{2} + 220 a^{5} b^{6} x^{3}} \]

[In]

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

[Out]

27*a**(35/3)*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) -
27*a**(35/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 63*a**(32/3)*b*x*(1
+ b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a**(32/3)*b*
x/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 42*a**(29/3)*b**2*x**2*(1 + b*
x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 81*a**(29/3)*b**2*x*
*2/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 78*a**(26/3)*b**3*x**3*(1 + b
*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) - 27*a**(26/3)*b**3*x
**3/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 207*a**(23/3)*b**4*x**4*(1 +
b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 195*a**(20/3)*b**
5*x**5*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3) + 60*a**
(17/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(220*a**8*b**3 + 660*a**7*b**4*x + 660*a**6*b**5*x**2 + 220*a**5*b**6*x**3
)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {11}{3}}}{11 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}} a}{4 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{2}}{5 \, b^{3}} \]

[In]

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

[Out]

3/11*(b*x + a)^(11/3)/b^3 - 3/4*(b*x + a)^(8/3)*a/b^3 + 3/5*(b*x + a)^(5/3)*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 = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {3 \, {\left (\frac {11 \, {\left (5 \, {\left (b x + a\right )}^{\frac {8}{3}} - 16 \, {\left (b x + a\right )}^{\frac {5}{3}} a + 20 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2}\right )} a}{b^{2}} + \frac {40 \, {\left (b x + a\right )}^{\frac {11}{3}} - 165 \, {\left (b x + a\right )}^{\frac {8}{3}} a + 264 \, {\left (b x + a\right )}^{\frac {5}{3}} a^{2} - 220 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{3}}{b^{2}}\right )}}{440 \, b} \]

[In]

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

[Out]

3/440*(11*(5*(b*x + a)^(8/3) - 16*(b*x + a)^(5/3)*a + 20*(b*x + a)^(2/3)*a^2)*a/b^2 + (40*(b*x + a)^(11/3) - 1
65*(b*x + a)^(8/3)*a + 264*(b*x + a)^(5/3)*a^2 - 220*(b*x + a)^(2/3)*a^3)/b^2)/b

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int x^2 (a+b x)^{2/3} \, dx=\frac {60\,{\left (a+b\,x\right )}^{11/3}-165\,a\,{\left (a+b\,x\right )}^{8/3}+132\,a^2\,{\left (a+b\,x\right )}^{5/3}}{220\,b^3} \]

[In]

int(x^2*(a + b*x)^(2/3),x)

[Out]

(60*(a + b*x)^(11/3) - 165*a*(a + b*x)^(8/3) + 132*a^2*(a + b*x)^(5/3))/(220*b^3)