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

Optimal. Leaf size=16 \[ \frac {x^2}{b \sqrt {b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, 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, 8} \[ \frac {x^2}{b \sqrt {b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.81 \[ \frac {x^4}{\left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^4/(b*x^2)^(3/2)

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fricas [A]  time = 0.75, size = 11, normalized size = 0.69 \[ \frac {\sqrt {b x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 11, normalized size = 0.69 \[ \frac {\sqrt {b x^{2}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 12, normalized size = 0.75 \[ \frac {x^{4}}{\left (b \,x^{2}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4/(b*x^2)^(3/2)

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maxima [A]  time = 1.30, size = 14, normalized size = 0.88 \[ \frac {x^{2}}{\sqrt {b x^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.99, size = 6, normalized size = 0.38 \[ \frac {\relax |x|}{b^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.51, size = 14, normalized size = 0.88 \[ \frac {x^{4}}{b^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/(b**(3/2)*(x**2)**(3/2))

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