3.1.77 \(\int \sqrt {b x^3} \, dx\) [77]

Optimal. Leaf size=14 \[ \frac {2}{5} x \sqrt {b x^3} \]

[Out]

2/5*x*(b*x^3)^(1/2)

________________________________________________________________________________________

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 {2}{5} x \sqrt {b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^3],x]

[Out]

(2*x*Sqrt[b*x^3])/5

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^3} \, dx &=\frac {\sqrt {b x^3} \int x^{3/2} \, dx}{x^{3/2}}\\ &=\frac {2}{5} x \sqrt {b x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {2}{5} x \sqrt {b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^3],x]

[Out]

(2*x*Sqrt[b*x^3])/5

________________________________________________________________________________________

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

method	result	size
gosper	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
default	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
trager	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)
risch	\(\frac {2 x \sqrt {b \,x^{3}}}{5}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/5*x*(b*x^3)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 10, normalized size = 0.71 \begin {gather*} \frac {2}{5} \, \sqrt {b x^{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/2),x, algorithm="maxima")

[Out]

2/5*sqrt(b*x^3)*x

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 10, normalized size = 0.71 \begin {gather*} \frac {2}{5} \, \sqrt {b x^{3}} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/2),x, algorithm="fricas")

[Out]

2/5*sqrt(b*x^3)*x

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 x \sqrt {b x^{3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3)**(1/2),x)

[Out]

2*x*sqrt(b*x**3)/5

________________________________________________________________________________________

Giac [A]
time = 2.09, size = 12, normalized size = 0.86 \begin {gather*} \frac {2}{5} \, \sqrt {b x} x^{2} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3)^(1/2),x, algorithm="giac")

[Out]

2/5*sqrt(b*x)*x^2*sgn(x)

________________________________________________________________________________________

Mupad [B]
time = 1.36, size = 10, normalized size = 0.71 \begin {gather*} \frac {2\,x\,\sqrt {b\,x^3}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*(b*x^3)^(1/2))/5

________________________________________________________________________________________