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

Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{5/3}}{5 b^3}+\frac{3 (a+b x)^{11/3}}{11 b^3}-\frac{3 a (a+b x)^{8/3}}{4 b^3} \]

[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)

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Rubi [A]  time = 0.0126484, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{3 a^2 (a+b x)^{5/3}}{5 b^3}+\frac{3 (a+b x)^{11/3}}{11 b^3}-\frac{3 a (a+b x)^{8/3}}{4 b^3} \]

Antiderivative was successfully verified.

[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 43

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*} \int x^2 (a+b x)^{2/3} \, dx &=\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]  time = 0.0376232, size = 35, normalized size = 0.66 \[ \frac{3 (a+b x)^{5/3} \left (9 a^2-15 a b x+20 b^2 x^2\right )}{220 b^3} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.6 \begin{align*}{\frac{60\,{b}^{2}{x}^{2}-45\,abx+27\,{a}^{2}}{220\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07781, size = 55, normalized size = 1.04 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.74525, size = 100, normalized size = 1.89 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [B]  time = 2.65505, size = 666, normalized size = 12.57 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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Giac [A]  time = 1.20521, size = 50, normalized size = 0.94 \begin{align*} \frac{3 \,{\left (20 \,{\left (b x + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x + a\right )}^{\frac{5}{3}} a^{2}\right )}}{220 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/220*(20*(b*x + a)^(11/3) - 55*(b*x + a)^(8/3)*a + 44*(b*x + a)^(5/3)*a^2)/b^3