3.1.22 \(\int (x^2)^{5/2} \, dx\) [22]

Optimal. Leaf size=14 \[ \frac {1}{6} x^5 \sqrt {x^2} \]

[Out]

1/6*x^5*(x^2)^(1/2)

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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 = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \begin {gather*} \frac {1}{6} x^5 \sqrt {x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2)^(5/2),x]

[Out]

(x^5*Sqrt[x^2])/6

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 (x^2\right )^{5/2} \, dx &=\frac {\sqrt {x^2} \int x^5 \, dx}{x}\\ &=\frac {1}{6} x^5 \sqrt {x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{6} x \left (x^2\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2)^(5/2),x]

[Out]

(x*(x^2)^(5/2))/6

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Maple [A]
time = 0.01, size = 9, normalized size = 0.64

method	result	size
gosper	\(\frac {x \left (x^{2}\right )^{\frac {5}{2}}}{6}\)	\(9\)
default	\(\frac {x \left (x^{2}\right )^{\frac {5}{2}}}{6}\)	\(9\)
risch	\(\frac {x^{5} \sqrt {x^{2}}}{6}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*x*(x^2)^(5/2)

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Maxima [A]
time = 0.30, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{6} \, {\left (x^{2}\right )}^{\frac {5}{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(x^2)^(5/2)*x

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Fricas [A]
time = 0.34, size = 5, normalized size = 0.36 \begin {gather*} \frac {1}{6} \, x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2)^(5/2),x, algorithm="fricas")

[Out]

1/6*x^6

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Sympy [A]
time = 0.01, size = 3, normalized size = 0.21 \begin {gather*} \frac {x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2)**(5/2),x)

[Out]

x**6/6

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Giac [A]
time = 1.51, size = 7, normalized size = 0.50 \begin {gather*} \frac {1}{6} \, x^{6} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2)^(5/2),x, algorithm="giac")

[Out]

1/6*x^6*sgn(x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int {\left (x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2)^(5/2),x)

[Out]

int((x^2)^(5/2), x)

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