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

Optimal. Leaf size=53 \[ \frac{3 a^2 (a+b x)^{7/3}}{7 b^3}+\frac{3 (a+b x)^{13/3}}{13 b^3}-\frac{3 a (a+b x)^{10/3}}{5 b^3} \]

[Out]

(3*a^2*(a + b*x)^(7/3))/(7*b^3) - (3*a*(a + b*x)^(10/3))/(5*b^3) + (3*(a + b*x)^(13/3))/(13*b^3)

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Rubi [A]  time = 0.0128579, 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} \[ \frac{3 a^2 (a+b x)^{7/3}}{7 b^3}+\frac{3 (a+b x)^{13/3}}{13 b^3}-\frac{3 a (a+b x)^{10/3}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(4/3),x]

[Out]

(3*a^2*(a + b*x)^(7/3))/(7*b^3) - (3*a*(a + b*x)^(10/3))/(5*b^3) + (3*(a + b*x)^(13/3))/(13*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)^{4/3} \, dx &=\int \left (\frac{a^2 (a+b x)^{4/3}}{b^2}-\frac{2 a (a+b x)^{7/3}}{b^2}+\frac{(a+b x)^{10/3}}{b^2}\right ) \, dx\\ &=\frac{3 a^2 (a+b x)^{7/3}}{7 b^3}-\frac{3 a (a+b x)^{10/3}}{5 b^3}+\frac{3 (a+b x)^{13/3}}{13 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0457376, size = 35, normalized size = 0.66 \[ \frac{3 (a+b x)^{7/3} \left (9 a^2-21 a b x+35 b^2 x^2\right )}{455 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3)*(9*a^2 - 21*a*b*x + 35*b^2*x^2))/(455*b^3)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.6 \begin{align*}{\frac{105\,{b}^{2}{x}^{2}-63\,abx+27\,{a}^{2}}{455\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/455*(b*x+a)^(7/3)*(35*b^2*x^2-21*a*b*x+9*a^2)/b^3

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Maxima [A]  time = 1.01368, size = 55, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{13}{3}}}{13 \, b^{3}} - \frac{3 \,{\left (b x + a\right )}^{\frac{10}{3}} a}{5 \, b^{3}} + \frac{3 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{2}}{7 \, 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]

3/13*(b*x + a)^(13/3)/b^3 - 3/5*(b*x + a)^(10/3)*a/b^3 + 3/7*(b*x + a)^(7/3)*a^2/b^3

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Fricas [A]  time = 1.47854, size = 123, normalized size = 2.32 \begin{align*} \frac{3 \,{\left (35 \, b^{4} x^{4} + 49 \, a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a^{3} b x + 9 \, a^{4}\right )}{\left (b x + a\right )}^{\frac{1}{3}}}{455 \, 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="fricas")

[Out]

3/455*(35*b^4*x^4 + 49*a*b^3*x^3 + 2*a^2*b^2*x^2 - 3*a^3*b*x + 9*a^4)*(b*x + a)^(1/3)/b^3

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Sympy [B]  time = 2.97957, size = 733, normalized size = 13.83 \begin{align*} \frac{27 a^{\frac{37}{3}} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{37}{3}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{72 a^{\frac{34}{3}} b x \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{34}{3}} b x}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{60 a^{\frac{31}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac{81 a^{\frac{31}{3}} b^{2} x^{2}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{165 a^{\frac{28}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac{27 a^{\frac{28}{3}} b^{3} x^{3}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{555 a^{\frac{25}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{762 a^{\frac{22}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{462 a^{\frac{19}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac{105 a^{\frac{16}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac{b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 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**(37/3)*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3)
- 27*a**(37/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 72*a**(34/3)*b*
x*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 81*a**(34
/3)*b*x/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 60*a**(31/3)*b**2*x**2
*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 81*a**(31/
3)*b**2*x**2/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 165*a**(28/3)*b**
3*x**3*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) - 27*a
**(28/3)*b**3*x**3/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3) + 555*a**(25/
3)*b**4*x**4*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**5*b**6*x**3)
+ 762*a**(22/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x**2 + 455*a**
5*b**6*x**3) + 462*a**(19/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 1365*a**6*b**5*x
**2 + 455*a**5*b**6*x**3) + 105*a**(16/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(455*a**8*b**3 + 1365*a**7*b**4*x + 136
5*a**6*b**5*x**2 + 455*a**5*b**6*x**3)

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Giac [B]  time = 1.18807, size = 124, normalized size = 2.34 \begin{align*} \frac{3 \,{\left (\frac{13 \,{\left (14 \,{\left (b x + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{2}\right )} a}{b^{2}} + \frac{140 \,{\left (b x + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x + a\right )}^{\frac{4}{3}} a^{3}}{b^{2}}\right )}}{1820 \, b} \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]

3/1820*(13*(14*(b*x + a)^(10/3) - 40*(b*x + a)^(7/3)*a + 35*(b*x + a)^(4/3)*a^2)*a/b^2 + (140*(b*x + a)^(13/3)
- 546*(b*x + a)^(10/3)*a + 780*(b*x + a)^(7/3)*a^2 - 455*(b*x + a)^(4/3)*a^3)/b^2)/b