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

Optimal. Leaf size=17 \[ \frac {1}{6} b x^5 \sqrt {b x^2} \]

[Out]

1/6*b*x^5*(b*x^2)^(1/2)

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac {1}{6} b x^5 \sqrt {b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^5*Sqrt[b*x^2])/6

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^2 \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^5 \, dx}{x}\\ &=\frac {1}{6} b x^5 \sqrt {b x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {1}{6} x^3 \left (b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 13, normalized size = 0.76 \[ \frac {1}{6} \, \sqrt {b x^{2}} b x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(b*x^2)*b*x^5

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giac [A]  time = 0.15, size = 10, normalized size = 0.59 \[ \frac {1}{6} \, b^{\frac {3}{2}} x^{6} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.76 \[ \frac {\left (b \,x^{2}\right )^{\frac {3}{2}} x^{3}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 13, normalized size = 0.76 \[ \frac {\left (b x^{2}\right )^{\frac {5}{2}} x}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2)^(5/2)*x/b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int x^2\,{\left (b\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(b*x^2)^(3/2), x)

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sympy [A]  time = 0.69, size = 15, normalized size = 0.88 \[ \frac {b^{\frac {3}{2}} x^{3} \left (x^{2}\right )^{\frac {3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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