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]

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

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Rubi [A]  time = 0.0016268, antiderivative size = 17, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0016987, 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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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{6} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28032, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{6} \, \sqrt{b x^{2}} b x^{5} \end{align*}

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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Sympy [A]  time = 0.608381, size = 15, normalized size = 0.88 \begin{align*} \frac{b^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \end{align*}

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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Giac [A]  time = 1.19117, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{6} \, b^{\frac{3}{2}} x^{6} \mathrm{sgn}\left (x\right ) \end{align*}

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)