3.1.78 \(\int \sqrt {b x^2} \, dx\) [78]

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

[Out]

1/2*x*(b*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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} \frac {1}{2} x \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[b*x^2])/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 \sqrt {b x^2} \, dx &=\frac {\sqrt {b x^2} \int x \, dx}{x}\\ &=\frac {1}{2} x \sqrt {b x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.00, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
default	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
risch	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
trager	\(\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.71 \begin {gather*} \frac {x \sqrt {b x^{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.39, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {b} x^{2} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.00, size = 8, normalized size = 0.57 \begin {gather*} \frac {\sqrt {b}\,x\,\left |x\right |}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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