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*Sqrt[b*x^2])

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Rubi [A]  time = 0.0014285, antiderivative size = 16, 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} \[ \frac{x^2}{b \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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{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*}

Mathematica [A]  time = 0.0010927, 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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Maple [A]  time = 0., size = 12, normalized size = 0.8 \begin{align*}{{x}^{4} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 1.01686, size = 19, normalized size = 1.19 \begin{align*} \frac{x^{2}}{\sqrt{b x^{2}} b} \end{align*}

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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Fricas [A]  time = 1.43097, size = 23, normalized size = 1.44 \begin{align*} \frac{\sqrt{b x^{2}}}{b^{2}} \end{align*}

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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Sympy [A]  time = 0.658561, size = 14, normalized size = 0.88 \begin{align*} \frac{x^{4}}{b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

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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Giac [A]  time = 1.13824, size = 15, normalized size = 0.94 \begin{align*} \frac{\sqrt{b x^{2}}}{b^{2}} \end{align*}

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