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

Optimal. Leaf size=51 \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0119599, antiderivative size = 51, 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} \[ \frac{3 a^2 \sqrt [3]{a+b x}}{b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^(2/3),x]

[Out]

(3*a^2*(a + b*x)^(1/3))/b^3 - (3*a*(a + b*x)^(4/3))/(2*b^3) + (3*(a + b*x)^(7/3))/(7*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}{(a+b x)^{2/3}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{2/3}}-\frac{2 a \sqrt [3]{a+b x}}{b^2}+\frac{(a+b x)^{4/3}}{b^2}\right ) \, dx\\ &=\frac{3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac{3 a (a+b x)^{4/3}}{2 b^3}+\frac{3 (a+b x)^{7/3}}{7 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0884474, size = 35, normalized size = 0.69 \[ \frac{3 \sqrt [3]{a+b x} \left (9 a^2-3 a b x+2 b^2 x^2\right )}{14 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^(2/3),x]

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}-9\,abx+27\,{a}^{2}}{14\,{b}^{3}}\sqrt [3]{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06806, size = 55, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}}}{7 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{4}{3}} a}{2 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2}}{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]

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

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Fricas [A]  time = 1.57083, size = 76, normalized size = 1.49 \begin{align*} \frac{3 \,{\left (2 \, b^{2} x^{2} - 3 \, a b x + 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{14 \, 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]

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

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Sympy [B]  time = 2.45954, size = 600, normalized size = 11.76 \begin{align*} \frac{27 a^{\frac{31}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{31}{3}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{72 a^{\frac{28}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{28}{3}} b x}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{25}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{25}{3}} b^{2} x^{2}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{18 a^{\frac{22}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{22}{3}} b^{3} x^{3}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{9 a^{\frac{19}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 a^{5} b^{6} x^{3}} + \frac{6 a^{\frac{16}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac{b x}{a}}}{14 a^{8} b^{3} + 42 a^{7} b^{4} x + 42 a^{6} b^{5} x^{2} + 14 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**(31/3)*(1 + b*x/a)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) - 27*a
**(31/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 72*a**(28/3)*b*x*(1 + b*x/a
)**(1/3)/(14*a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) - 81*a**(28/3)*b*x/(14*a**8*b
**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 60*a**(25/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(14*
a**8*b**3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) - 81*a**(25/3)*b**2*x**2/(14*a**8*b**3 + 4
2*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 18*a**(22/3)*b**3*x**3*(1 + b*x/a)**(1/3)/(14*a**8*b*
*3 + 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) - 27*a**(22/3)*b**3*x**3/(14*a**8*b**3 + 42*a**7*
b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 9*a**(19/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(14*a**8*b**3 + 42*
a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3) + 6*a**(16/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(14*a**8*b**3
+ 42*a**7*b**4*x + 42*a**6*b**5*x**2 + 14*a**5*b**6*x**3)

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Giac [A]  time = 1.19898, size = 50, normalized size = 0.98 \begin{align*} \frac{3 \,{\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 )}}{14 \, 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]

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