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

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

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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 {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*}

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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

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IntegrateAlgebraic [A]  time = 0.02, size = 19, normalized size = 1.36 \begin {gather*} \frac {2 \left (b x^3\right )^{5/2}}{5 b^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, 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

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giac [A]  time = 0.17, size = 12, normalized size = 0.86 \begin {gather*} \frac {2}{5} \, \sqrt {b x} x^{2} \mathrm {sgn}\relax (x) \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)

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maple [A]  time = 0.00, size = 11, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {b \,x^{3}}\, x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, 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="maxima")

[Out]

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

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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

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sympy [A]  time = 0.17, size = 15, normalized size = 1.07 \begin {gather*} \frac {2 \sqrt {b} x \sqrt {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*sqrt(b)*x*sqrt(x**3)/5

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