3.414 \(\int \frac{x^2}{(a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]

[Out]

(-3*a^2)/(b^3*(a + b*x)^(1/3)) - (3*a*(a + b*x)^(2/3))/b^3 + (3*(a + b*x)^(5/3))/(5*b^3)

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Rubi [A]  time = 0.0133071, antiderivative size = 49, 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} \[ -\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^(4/3),x]

[Out]

(-3*a^2)/(b^3*(a + b*x)^(1/3)) - (3*a*(a + b*x)^(2/3))/b^3 + (3*(a + b*x)^(5/3))/(5*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)^{4/3}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{4/3}}-\frac{2 a}{b^2 \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b^2}\right ) \, dx\\ &=-\frac{3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac{3 a (a+b x)^{2/3}}{b^3}+\frac{3 (a+b x)^{5/3}}{5 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0900145, size = 34, normalized size = 0.69 \[ \frac{3 \left (-9 a^2-3 a b x+b^2 x^2\right )}{5 b^3 \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^(4/3),x]

[Out]

(3*(-9*a^2 - 3*a*b*x + b^2*x^2))/(5*b^3*(a + b*x)^(1/3))

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Maple [A]  time = 0.005, size = 32, normalized size = 0.7 \begin{align*} -{\frac{-3\,{b}^{2}{x}^{2}+9\,abx+27\,{a}^{2}}{5\,{b}^{3}}{\frac{1}{\sqrt [3]{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(4/3),x)

[Out]

-3/5/(b*x+a)^(1/3)*(-b^2*x^2+3*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.06022, size = 55, normalized size = 1.12 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

3/5*(b*x + a)^(5/3)/b^3 - 3*(b*x + a)^(2/3)*a/b^3 - 3*a^2/((b*x + a)^(1/3)*b^3)

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Fricas [A]  time = 1.52898, size = 88, normalized size = 1.8 \begin{align*} \frac{3 \,{\left (b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{5 \,{\left (b^{4} x + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(4/3),x, algorithm="fricas")

[Out]

3/5*(b^2*x^2 - 3*a*b*x - 9*a^2)*(b*x + a)^(2/3)/(b^4*x + a*b^3)

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Sympy [B]  time = 2.52779, size = 534, normalized size = 10.9 \begin{align*} - \frac{27 a^{\frac{29}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{29}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{63 a^{\frac{26}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{26}{3}} b x}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{42 a^{\frac{23}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{81 a^{\frac{23}{3}} b^{2} x^{2}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} - \frac{3 a^{\frac{20}{3}} b^{3} x^{3} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{27 a^{\frac{20}{3}} b^{3} x^{3}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 a^{5} b^{6} x^{3}} + \frac{3 a^{\frac{17}{3}} b^{4} x^{4} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{5 a^{8} b^{3} + 15 a^{7} b^{4} x + 15 a^{6} b^{5} x^{2} + 5 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)**(4/3),x)

[Out]

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

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Giac [A]  time = 1.17738, size = 62, normalized size = 1.27 \begin{align*} -\frac{3 \, a^{2}}{{\left (b x + a\right )}^{\frac{1}{3}} b^{3}} + \frac{3 \,{\left ({\left (b x + a\right )}^{\frac{5}{3}} b^{12} - 5 \,{\left (b x + a\right )}^{\frac{2}{3}} a b^{12}\right )}}{5 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(4/3),x, algorithm="giac")

[Out]

-3*a^2/((b*x + a)^(1/3)*b^3) + 3/5*((b*x + a)^(5/3)*b^12 - 5*(b*x + a)^(2/3)*a*b^12)/b^15