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

Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{2/3}}{2 b^3}+\frac{3 (a+b x)^{8/3}}{8 b^3}-\frac{6 a (a+b x)^{5/3}}{5 b^3} \]

[Out]

(3*a^2*(a + b*x)^(2/3))/(2*b^3) - (6*a*(a + b*x)^(5/3))/(5*b^3) + (3*(a + b*x)^(8/3))/(8*b^3)

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Rubi [A]  time = 0.0124392, 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} \[ \frac{3 a^2 (a+b x)^{2/3}}{2 b^3}+\frac{3 (a+b x)^{8/3}}{8 b^3}-\frac{6 a (a+b x)^{5/3}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^(1/3),x]

[Out]

(3*a^2*(a + b*x)^(2/3))/(2*b^3) - (6*a*(a + b*x)^(5/3))/(5*b^3) + (3*(a + b*x)^(8/3))/(8*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 \frac{x^2}{\sqrt [3]{a+b x}} \, dx &=\int \left (\frac{a^2}{b^2 \sqrt [3]{a+b x}}-\frac{2 a (a+b x)^{2/3}}{b^2}+\frac{(a+b x)^{5/3}}{b^2}\right ) \, dx\\ &=\frac{3 a^2 (a+b x)^{2/3}}{2 b^3}-\frac{6 a (a+b x)^{5/3}}{5 b^3}+\frac{3 (a+b x)^{8/3}}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0333041, size = 35, normalized size = 0.66 \[ \frac{3 (a+b x)^{2/3} \left (9 a^2-6 a b x+5 b^2 x^2\right )}{40 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(2/3)*(9*a^2 - 6*a*b*x + 5*b^2*x^2))/(40*b^3)

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Maple [A]  time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{15\,{b}^{2}{x}^{2}-18\,abx+27\,{a}^{2}}{40\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(b*x+a)^(2/3)*(5*b^2*x^2-6*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.00102, size = 55, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{8}{3}}}{8 \, b^{3}} - \frac{6 \,{\left (b x + a\right )}^{\frac{5}{3}} a}{5 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x + a)^(8/3)/b^3 - 6/5*(b*x + a)^(5/3)*a/b^3 + 3/2*(b*x + a)^(2/3)*a^2/b^3

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Fricas [A]  time = 1.49519, size = 76, normalized size = 1.43 \begin{align*} \frac{3 \,{\left (5 \, b^{2} x^{2} - 6 \, a b x + 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{40 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(5*b^2*x^2 - 6*a*b*x + 9*a^2)*(b*x + a)^(2/3)/b^3

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Sympy [B]  time = 2.18696, size = 600, normalized size = 11.32 \begin{align*} \frac{27 a^{\frac{32}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{32}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{63 a^{\frac{29}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{29}{3}} b x}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{42 a^{\frac{26}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{26}{3}} b^{2} x^{2}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{18 a^{\frac{23}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{23}{3}} b^{3} x^{3}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{20}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 a^{5} b^{6} x^{3}} + \frac{15 a^{\frac{17}{3}} b^{5} x^{5} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{40 a^{8} b^{3} + 120 a^{7} b^{4} x + 120 a^{6} b^{5} x^{2} + 40 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)**(1/3),x)

[Out]

27*a**(32/3)*(1 + b*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 27
*a**(32/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 63*a**(29/3)*b*x*(1 + b
*x/a)**(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 81*a**(29/3)*b*x/(40*
a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 42*a**(26/3)*b**2*x**2*(1 + b*x/a)**(2
/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 81*a**(26/3)*b**2*x**2/(40*a**
8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 18*a**(23/3)*b**3*x**3*(1 + b*x/a)**(2/3)
/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) - 27*a**(23/3)*b**3*x**3/(40*a**8*b
**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 27*a**(20/3)*b**4*x**4*(1 + b*x/a)**(2/3)/(4
0*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3) + 15*a**(17/3)*b**5*x**5*(1 + b*x/a)**
(2/3)/(40*a**8*b**3 + 120*a**7*b**4*x + 120*a**6*b**5*x**2 + 40*a**5*b**6*x**3)

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Giac [A]  time = 1.13827, size = 50, normalized size = 0.94 \begin{align*} \frac{3 \,{\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 )}}{40 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(5*(b*x + a)^(8/3) - 16*(b*x + a)^(5/3)*a + 20*(b*x + a)^(2/3)*a^2)/b^3