3.61 \(\int \frac{1}{x^2 (b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{6 b^2 x^5 \sqrt{b x^2}} \]

[Out]

-1/(6*b^2*x^5*Sqrt[b*x^2])

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Rubi [A]  time = 0.0018788, 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} \[ -\frac{1}{6 b^2 x^5 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(b*x^2)^(5/2)),x]

[Out]

-1/(6*b^2*x^5*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 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{1}{x^2 \left (b x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{1}{x^7} \, dx}{b^2 \sqrt{b x^2}}\\ &=-\frac{1}{6 b^2 x^5 \sqrt{b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.001991, size = 15, normalized size = 0.79 \[ -\frac{b x}{6 \left (b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(b*x^2)^(5/2)),x]

[Out]

-(b*x)/(6*(b*x^2)^(7/2))

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Maple [A]  time = 0.001, size = 13, normalized size = 0.7 \begin{align*} -{\frac{1}{6\,x} \left ( b{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^2)^(5/2),x)

[Out]

-1/6/x/(b*x^2)^(5/2)

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Maxima [A]  time = 1.0052, size = 11, normalized size = 0.58 \begin{align*} -\frac{1}{6 \, b^{\frac{5}{2}} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2)^(5/2),x, algorithm="maxima")

[Out]

-1/6/(b^(5/2)*x^6)

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Fricas [A]  time = 1.4332, size = 38, normalized size = 2. \begin{align*} -\frac{\sqrt{b x^{2}}}{6 \, b^{3} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2)^(5/2),x, algorithm="fricas")

[Out]

-1/6*sqrt(b*x^2)/(b^3*x^7)

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Sympy [A]  time = 1.27276, size = 17, normalized size = 0.89 \begin{align*} - \frac{1}{6 b^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2)**(5/2),x)

[Out]

-1/(6*b**(5/2)*x*(x**2)**(5/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x