3.2.10 \(\int (b x^2)^{2/3} \, dx\) [110]

Optimal. Leaf size=14 \[ \frac {3}{7} x \left (b x^2\right )^{2/3} \]

[Out]

3/7*x*(b*x^2)^(2/3)

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} \frac {3}{7} x \left (b x^2\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(2/3),x]

[Out]

(3*x*(b*x^2)^(2/3))/7

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (b x^2\right )^{2/3} \, dx &=\frac {\left (b x^2\right )^{2/3} \int x^{4/3} \, dx}{x^{4/3}}\\ &=\frac {3}{7} x \left (b x^2\right )^{2/3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{7} x \left (b x^2\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(2/3),x]

[Out]

(3*x*(b*x^2)^(2/3))/7

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Maple [A]
time = 0.03, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {3 x \left (b \,x^{2}\right )^{\frac {2}{3}}}{7}\)	\(11\)
trager	\(\frac {3 x \left (b \,x^{2}\right )^{\frac {2}{3}}}{7}\)	\(11\)
risch	\(\frac {3 b \,x^{3}}{7 \left (b \,x^{2}\right )^{\frac {1}{3}}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/7*x*(b*x^2)^(2/3)

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Maxima [A]
time = 0.30, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{7} \, \left (b x^{2}\right )^{\frac {2}{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(b*x^2)^(2/3)*x

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Fricas [A]
time = 0.34, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{7} \, \left (b x^{2}\right )^{\frac {2}{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(b*x^2)^(2/3)*x

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Sympy [A]
time = 0.09, size = 12, normalized size = 0.86 \begin {gather*} \frac {3 x \left (b x^{2}\right )^{\frac {2}{3}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*(b*x**2)**(2/3)/7

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Giac [A]
time = 1.77, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{7} \, \left (b x^{2}\right )^{\frac {2}{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*(b*x^2)^(2/3)*x

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Mupad [B]
time = 1.00, size = 11, normalized size = 0.79 \begin {gather*} \frac {3\,b^{2/3}\,x\,{\left (x^4\right )}^{1/3}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*b^(2/3)*x*(x^4)^(1/3))/7

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