3.666 \(\int x^{5/3} (a+b x)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{9}{11} a^2 b x^{11/3}+\frac{3}{8} a^3 x^{8/3}+\frac{9}{14} a b^2 x^{14/3}+\frac{3}{17} b^3 x^{17/3} \]

[Out]

(3*a^3*x^(8/3))/8 + (9*a^2*b*x^(11/3))/11 + (9*a*b^2*x^(14/3))/14 + (3*b^3*x^(17/3))/17

________________________________________________________________________________________

Rubi [A]  time = 0.0104423, 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} \[ \frac{9}{11} a^2 b x^{11/3}+\frac{3}{8} a^3 x^{8/3}+\frac{9}{14} a b^2 x^{14/3}+\frac{3}{17} b^3 x^{17/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/3)*(a + b*x)^3,x]

[Out]

(3*a^3*x^(8/3))/8 + (9*a^2*b*x^(11/3))/11 + (9*a*b^2*x^(14/3))/14 + (3*b^3*x^(17/3))/17

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^{5/3} (a+b x)^3 \, dx &=\int \left (a^3 x^{5/3}+3 a^2 b x^{8/3}+3 a b^2 x^{11/3}+b^3 x^{14/3}\right ) \, dx\\ &=\frac{3}{8} a^3 x^{8/3}+\frac{9}{11} a^2 b x^{11/3}+\frac{9}{14} a b^2 x^{14/3}+\frac{3}{17} b^3 x^{17/3}\\ \end{align*}

Mathematica [A]  time = 0.0117228, size = 39, normalized size = 0.76 \[ \frac{3 x^{8/3} \left (2856 a^2 b x+1309 a^3+2244 a b^2 x^2+616 b^3 x^3\right )}{10472} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/3)*(a + b*x)^3,x]

[Out]

(3*x^(8/3)*(1309*a^3 + 2856*a^2*b*x + 2244*a*b^2*x^2 + 616*b^3*x^3))/10472

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 36, normalized size = 0.7 \begin{align*}{\frac{1848\,{b}^{3}{x}^{3}+6732\,a{b}^{2}{x}^{2}+8568\,{a}^{2}bx+3927\,{a}^{3}}{10472}{x}^{{\frac{8}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)*(b*x+a)^3,x)

[Out]

3/10472*x^(8/3)*(616*b^3*x^3+2244*a*b^2*x^2+2856*a^2*b*x+1309*a^3)

________________________________________________________________________________________

Maxima [A]  time = 1.42024, size = 47, normalized size = 0.92 \begin{align*} \frac{3}{17} \, b^{3} x^{\frac{17}{3}} + \frac{9}{14} \, a b^{2} x^{\frac{14}{3}} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{3}{8} \, a^{3} x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a)^3,x, algorithm="maxima")

[Out]

3/17*b^3*x^(17/3) + 9/14*a*b^2*x^(14/3) + 9/11*a^2*b*x^(11/3) + 3/8*a^3*x^(8/3)

________________________________________________________________________________________

Fricas [A]  time = 1.54968, size = 108, normalized size = 2.12 \begin{align*} \frac{3}{10472} \,{\left (616 \, b^{3} x^{5} + 2244 \, a b^{2} x^{4} + 2856 \, a^{2} b x^{3} + 1309 \, a^{3} x^{2}\right )} x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a)^3,x, algorithm="fricas")

[Out]

3/10472*(616*b^3*x^5 + 2244*a*b^2*x^4 + 2856*a^2*b*x^3 + 1309*a^3*x^2)*x^(2/3)

________________________________________________________________________________________

Sympy [A]  time = 7.31497, size = 49, normalized size = 0.96 \begin{align*} \frac{3 a^{3} x^{\frac{8}{3}}}{8} + \frac{9 a^{2} b x^{\frac{11}{3}}}{11} + \frac{9 a b^{2} x^{\frac{14}{3}}}{14} + \frac{3 b^{3} x^{\frac{17}{3}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/3)*(b*x+a)**3,x)

[Out]

3*a**3*x**(8/3)/8 + 9*a**2*b*x**(11/3)/11 + 9*a*b**2*x**(14/3)/14 + 3*b**3*x**(17/3)/17

________________________________________________________________________________________

Giac [A]  time = 1.07897, size = 47, normalized size = 0.92 \begin{align*} \frac{3}{17} \, b^{3} x^{\frac{17}{3}} + \frac{9}{14} \, a b^{2} x^{\frac{14}{3}} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{3}{8} \, a^{3} x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a)^3,x, algorithm="giac")

[Out]

3/17*b^3*x^(17/3) + 9/14*a*b^2*x^(14/3) + 9/11*a^2*b*x^(11/3) + 3/8*a^3*x^(8/3)