3.32 \(\int \frac{(b x^2)^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=19 \[ -\frac{b^2 \sqrt{b x^2}}{2 x^3} \]

[Out]

-(b^2*Sqrt[b*x^2])/(2*x^3)

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Rubi [A]  time = 0.0018092, antiderivative size = 19, 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} \[ -\frac{b^2 \sqrt{b x^2}}{2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^(5/2)/x^8,x]

[Out]

-(b^2*Sqrt[b*x^2])/(2*x^3)

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 )^{5/2}}{x^8} \, dx &=\frac{\left (b^2 \sqrt{b x^2}\right ) \int \frac{1}{x^3} \, dx}{x}\\ &=-\frac{b^2 \sqrt{b x^2}}{2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0012862, size = 16, normalized size = 0.84 \[ -\frac{\left (b x^2\right )^{5/2}}{2 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^(5/2)/x^8,x]

[Out]

-(b*x^2)^(5/2)/(2*x^7)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,{x}^{7}} \left ( b{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(5/2)/x^8,x)

[Out]

-1/2*(b*x^2)^(5/2)/x^7

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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)^(5/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.2748, size = 35, normalized size = 1.84 \begin{align*} -\frac{\sqrt{b x^{2}} b^{2}}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^8,x, algorithm="fricas")

[Out]

-1/2*sqrt(b*x^2)*b^2/x^3

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Sympy [A]  time = 2.7934, size = 17, normalized size = 0.89 \begin{align*} - \frac{b^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{2 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(5/2)/x**8,x)

[Out]

-b**(5/2)*(x**2)**(5/2)/(2*x**7)

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Giac [A]  time = 1.10933, size = 14, normalized size = 0.74 \begin{align*} -\frac{b^{\frac{5}{2}} \mathrm{sgn}\left (x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(5/2)/x^8,x, algorithm="giac")

[Out]

-1/2*b^(5/2)*sgn(x)/x^2