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

Optimal. Leaf size=11 \[ b \sqrt{b x^2} \]

[Out]

b*Sqrt[b*x^2]

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Rubi [A]  time = 0.0010735, antiderivative size = 11, 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, 8} \[ b \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*Sqrt[b*x^2]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0009498, size = 13, normalized size = 1.18 \[ \frac{\left (b x^2\right )^{3/2}}{x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 12, normalized size = 1.1 \begin{align*}{\frac{1}{{x}^{2}} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.21058, size = 20, normalized size = 1.82 \begin{align*} \sqrt{b x^{2}} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)*b

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Sympy [A]  time = 0.697533, size = 14, normalized size = 1.27 \begin{align*} \frac{b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15732, size = 9, normalized size = 0.82 \begin{align*} b^{\frac{3}{2}} x \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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