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

Optimal. Leaf size=16 \[ \frac{3 (a+b x)^{5/3}}{5 b} \]

[Out]

(3*(a + b*x)^(5/3))/(5*b)

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Rubi [A]  time = 0.0015048, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{3 (a+b x)^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3))/(5*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0041166, size = 16, normalized size = 1. \[ \frac{3 (a+b x)^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(2/3),x]

[Out]

(3*(a + b*x)^(5/3))/(5*b)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*}{\frac{3}{5\,b} \left ( bx+a \right ) ^{{\frac{5}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1062, size = 16, normalized size = 1. \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(b*x + a)^(5/3)/b

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Fricas [A]  time = 1.83823, size = 31, normalized size = 1.94 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(b*x + a)^(5/3)/b

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Sympy [A]  time = 0.073232, size = 12, normalized size = 0.75 \begin{align*} \frac{3 \left (a + b x\right )^{\frac{5}{3}}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(a + b*x)**(5/3)/(5*b)

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Giac [A]  time = 1.14559, size = 16, normalized size = 1. \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(b*x + a)^(5/3)/b