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

Optimal. Leaf size=21 \[ \frac{3}{5} a x^{5/3}+\frac{3}{8} b x^{8/3} \]

[Out]

(3*a*x^(5/3))/5 + (3*b*x^(8/3))/8

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Rubi [A]  time = 0.003643, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3}{5} a x^{5/3}+\frac{3}{8} b x^{8/3} \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)*(a + b*x),x]

[Out]

(3*a*x^(5/3))/5 + (3*b*x^(8/3))/8

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

Mathematica [A]  time = 0.0046576, size = 17, normalized size = 0.81 \[ \frac{3}{40} x^{5/3} (8 a+5 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)*(a + b*x),x]

[Out]

(3*x^(5/3)*(8*a + 5*b*x))/40

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{15\,bx+24\,a}{40}{x}^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*x^(5/3)*(5*b*x+8*a)

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Maxima [A]  time = 1.0489, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{8} \, b x^{\frac{8}{3}} + \frac{3}{5} \, a x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*b*x^(8/3) + 3/5*a*x^(5/3)

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Fricas [A]  time = 1.54449, size = 43, normalized size = 2.05 \begin{align*} \frac{3}{40} \,{\left (5 \, b x^{2} + 8 \, a x\right )} x^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40*(5*b*x^2 + 8*a*x)*x^(2/3)

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Sympy [A]  time = 0.485269, size = 19, normalized size = 0.9 \begin{align*} \frac{3 a x^{\frac{5}{3}}}{5} + \frac{3 b x^{\frac{8}{3}}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a*x**(5/3)/5 + 3*b*x**(8/3)/8

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Giac [A]  time = 1.05589, size = 18, normalized size = 0.86 \begin{align*} \frac{3}{8} \, b x^{\frac{8}{3}} + \frac{3}{5} \, a x^{\frac{5}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*b*x^(8/3) + 3/5*a*x^(5/3)