3.658 \(\int x^{5/3} (a+b x)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{3}{8} a^2 x^{8/3}+\frac{6}{11} a b x^{11/3}+\frac{3}{14} b^2 x^{14/3} \]

[Out]

(3*a^2*x^(8/3))/8 + (6*a*b*x^(11/3))/11 + (3*b^2*x^(14/3))/14

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Rubi [A]  time = 0.0067997, antiderivative size = 36, 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} \[ \frac{3}{8} a^2 x^{8/3}+\frac{6}{11} a b x^{11/3}+\frac{3}{14} b^2 x^{14/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/3)*(a + b*x)^2,x]

[Out]

(3*a^2*x^(8/3))/8 + (6*a*b*x^(11/3))/11 + (3*b^2*x^(14/3))/14

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)^2 \, dx &=\int \left (a^2 x^{5/3}+2 a b x^{8/3}+b^2 x^{11/3}\right ) \, dx\\ &=\frac{3}{8} a^2 x^{8/3}+\frac{6}{11} a b x^{11/3}+\frac{3}{14} b^2 x^{14/3}\\ \end{align*}

Mathematica [A]  time = 0.0083215, size = 28, normalized size = 0.78 \[ \frac{3}{616} x^{8/3} \left (77 a^2+112 a b x+44 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/3)*(a + b*x)^2,x]

[Out]

(3*x^(8/3)*(77*a^2 + 112*a*b*x + 44*b^2*x^2))/616

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Maple [A]  time = 0.003, size = 25, normalized size = 0.7 \begin{align*}{\frac{132\,{b}^{2}{x}^{2}+336\,abx+231\,{a}^{2}}{616}{x}^{{\frac{8}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)*(b*x+a)^2,x)

[Out]

3/616*x^(8/3)*(44*b^2*x^2+112*a*b*x+77*a^2)

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Maxima [A]  time = 1.00203, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{8} \, a^{2} x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a)^2,x, algorithm="maxima")

[Out]

3/14*b^2*x^(14/3) + 6/11*a*b*x^(11/3) + 3/8*a^2*x^(8/3)

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Fricas [A]  time = 1.46021, size = 74, normalized size = 2.06 \begin{align*} \frac{3}{616} \,{\left (44 \, b^{2} x^{4} + 112 \, a b x^{3} + 77 \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

3/616*(44*b^2*x^4 + 112*a*b*x^3 + 77*a^2*x^2)*x^(2/3)

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Sympy [A]  time = 4.0259, size = 34, normalized size = 0.94 \begin{align*} \frac{3 a^{2} x^{\frac{8}{3}}}{8} + \frac{6 a b x^{\frac{11}{3}}}{11} + \frac{3 b^{2} x^{\frac{14}{3}}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/3)*(b*x+a)**2,x)

[Out]

3*a**2*x**(8/3)/8 + 6*a*b*x**(11/3)/11 + 3*b**2*x**(14/3)/14

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Giac [A]  time = 1.07298, size = 32, normalized size = 0.89 \begin{align*} \frac{3}{14} \, b^{2} x^{\frac{14}{3}} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{8} \, a^{2} x^{\frac{8}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)*(b*x+a)^2,x, algorithm="giac")

[Out]

3/14*b^2*x^(14/3) + 6/11*a*b*x^(11/3) + 3/8*a^2*x^(8/3)