3.1.4 \(\int x \sqrt {b x^2} \, dx\) [4]

Optimal. Leaf size=16 \[ \frac {1}{3} x^2 \sqrt {b x^2} \]

[Out]

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

________________________________________________________________________________________

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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30} \begin {gather*} \frac {1}{3} x^2 \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 x \sqrt {b x^2} \, dx &=\frac {\sqrt {b x^2} \int x^2 \, dx}{x}\\ &=\frac {1}{3} x^2 \sqrt {b x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^2 \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \sqrt {b x^{2}} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^{2} \sqrt {b x^{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.90, size = 10, normalized size = 0.62 \begin {gather*} \frac {1}{3} \, \sqrt {b} x^{3} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(b)*x^3*sgn(x)

________________________________________________________________________________________

Mupad [B]
time = 1.02, size = 10, normalized size = 0.62 \begin {gather*} \frac {\sqrt {b}\,\sqrt {x^6}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^(1/2)*(x^6)^(1/2))/3

________________________________________________________________________________________