3.1.13 \(\int x (b x^2)^{3/2} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{5} b x^4 \sqrt {b x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, 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}{5} b x^4 \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b*x^2)^(3/2),x]

[Out]

(b*x^4*Sqrt[b*x^2])/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 x \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^4 \, dx}{x}\\ &=\frac {1}{5} b x^4 \sqrt {b x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {1}{5} x^2 \left (b x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b*x^2)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (b x^2\right )^{5/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x*(b*x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.95, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, \sqrt {b x^{2}} b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(b*x^2)*b*x^4

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giac [A]  time = 0.17, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{5} \, b^{\frac {3}{2}} x^{5} \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.76 \begin {gather*} \frac {\left (b \,x^{2}\right )^{\frac {3}{2}} x^{2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 12, normalized size = 0.71 \begin {gather*} \frac {\left (b x^{2}\right )^{\frac {5}{2}}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.96, size = 10, normalized size = 0.59 \begin {gather*} \frac {b^{3/2}\,\sqrt {x^{10}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^(3/2)*(x^10)^(1/2))/5

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sympy [A]  time = 0.51, size = 15, normalized size = 0.88 \begin {gather*} \frac {b^{\frac {3}{2}} x^{2} \left (x^{2}\right )^{\frac {3}{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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