3.1.39 \(\int \frac {x^2}{\sqrt {b x^2}} \, dx\) [39]

Optimal. Leaf size=16 \[ \frac {x^3}{2 \sqrt {b x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} \frac {x^3}{2 \sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[b*x^2],x]

[Out]

x^3/(2*Sqrt[b*x^2])

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 \frac {x^2}{\sqrt {b x^2}} \, dx &=\frac {x \int x \, dx}{\sqrt {b x^2}}\\ &=\frac {x^3}{2 \sqrt {b x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^3}{2 \sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[b*x^2],x]

[Out]

x^3/(2*Sqrt[b*x^2])

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Maple [A]
time = 0.02, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {x^{3}}{2 \sqrt {b \,x^{2}}}\)	\(13\)
default	\(\frac {x^{3}}{2 \sqrt {b \,x^{2}}}\)	\(13\)
risch	\(\frac {x^{3}}{2 \sqrt {b \,x^{2}}}\)	\(13\)
trager	\(\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 b x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 8, normalized size = 0.50 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*x^2/sqrt(b)

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Fricas [A]
time = 0.34, size = 13, normalized size = 0.81 \begin {gather*} \frac {\sqrt {b x^{2}} x}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(b*x^2)*x/b

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Sympy [A]
time = 0.17, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{3}}{2 \sqrt {b x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**2)**(1/2),x)

[Out]

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

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Giac [A]
time = 0.96, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {b} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x^2)^(1/2),x, algorithm="giac")

[Out]

1/2*x^2/(sqrt(b)*sgn(x))

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Mupad [B]
time = 0.99, size = 11, normalized size = 0.69 \begin {gather*} \frac {x\,\sqrt {x^2}}{2\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2)^(1/2),x)

[Out]

(x*(x^2)^(1/2))/(2*b^(1/2))

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