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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 = 0.02, 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)

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 53, normalized size = 1.00 \[ \frac {3 \, {\left (35 \, b^{4} x^{4} + 49 \, a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a^{3} b x + 9 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{455 \, b^{3}} \]

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

________________________________________________________________________________________

giac [B]  time = 0.98, size = 157, normalized size = 2.96 \[ \frac {3 \, {\left (\frac {65 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )} a^{2}}{b^{2}} + \frac {13 \, {\left (14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}\right )} a}{b^{2}} + \frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {13}{3}} - 182 \, {\left (b x + a\right )}^{\frac {10}{3}} a + 390 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2} - 455 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{3} + 455 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{4}\right )}}{b^{2}}\right )}}{910 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/910*(65*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 14*(b*x + a)^(1/3)*a^2)*a^2/b^2 + 13*(14*(b*x + a)^(10/3)
- 60*(b*x + a)^(7/3)*a + 105*(b*x + a)^(4/3)*a^2 - 140*(b*x + a)^(1/3)*a^3)*a/b^2 + 2*(35*(b*x + a)^(13/3) -
182*(b*x + a)^(10/3)*a + 390*(b*x + a)^(7/3)*a^2 - 455*(b*x + a)^(4/3)*a^3 + 455*(b*x + a)^(1/3)*a^4)/b^2)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.60 \[ \frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (35 b^{2} x^{2}-21 a b x +9 a^{2}\right )}{455 b^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 41, normalized size = 0.77 \[ \frac {3 \, {\left (b x + a\right )}^{\frac {13}{3}}}{13 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}} a}{5 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2}}{7 \, b^{3}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \[ \frac {105\,{\left (a+b\,x\right )}^{13/3}-273\,a\,{\left (a+b\,x\right )}^{10/3}+195\,a^2\,{\left (a+b\,x\right )}^{7/3}}{455\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(4/3),x)

[Out]

(105*(a + b*x)^(13/3) - 273*a*(a + b*x)^(10/3) + 195*a^2*(a + b*x)^(7/3))/(455*b^3)

________________________________________________________________________________________

sympy [B]  time = 2.16, size = 733, normalized size = 13.83 \[ \frac {27 a^{\frac {37}{3}} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {37}{3}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {72 a^{\frac {34}{3}} b x \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {34}{3}} b x}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {60 a^{\frac {31}{3}} b^{2} x^{2} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {81 a^{\frac {31}{3}} b^{2} x^{2}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {165 a^{\frac {28}{3}} b^{3} x^{3} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} - \frac {27 a^{\frac {28}{3}} b^{3} x^{3}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {555 a^{\frac {25}{3}} b^{4} x^{4} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {762 a^{\frac {22}{3}} b^{5} x^{5} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {462 a^{\frac {19}{3}} b^{6} x^{6} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} + \frac {105 a^{\frac {16}{3}} b^{7} x^{7} \sqrt [3]{1 + \frac {b x}{a}}}{455 a^{8} b^{3} + 1365 a^{7} b^{4} x + 1365 a^{6} b^{5} x^{2} + 455 a^{5} b^{6} x^{3}} \]

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)

________________________________________________________________________________________