3.19 \(\int \frac{(b x^2)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=15 \[ -\frac{b \sqrt{b x^2}}{x^2} \]

[Out]

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

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Rubi [A]  time = 0.0016574, antiderivative size = 15, 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} \[ -\frac{b \sqrt{b x^2}}{x^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(3/2)/x^5,x]

[Out]

-((b*Sqrt[b*x^2])/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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.001482, size = 14, normalized size = 0.93 \[ -\frac{\left (b x^2\right )^{3/2}}{x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(3/2)/x^5,x]

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{{x}^{4}} \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^5,x)

[Out]

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

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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^5,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.1409, size = 27, normalized size = 1.8 \begin{align*} -\frac{\sqrt{b x^{2}} b}{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.13479, size = 15, normalized size = 1. \begin{align*} - \frac{b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(3/2)/x**5,x)

[Out]

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

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Giac [A]  time = 1.12685, size = 14, normalized size = 0.93 \begin{align*} -\frac{b^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(3/2)/x^5,x, algorithm="giac")

[Out]

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