3.503 \(\int \frac {1}{x^2 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac {2 b x}{a^2 \sqrt {a+b x^2}}-\frac {1}{a x \sqrt {a+b x^2}} \]

[Out]

-1/a/x/(b*x^2+a)^(1/2)-2*b*x/a^2/(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 191} \[ -\frac {2 b x}{a^2 \sqrt {a+b x^2}}-\frac {1}{a x \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^(3/2)),x]

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \[ -\frac {a+2 b x^2}{a^2 x \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^(3/2)),x]

[Out]

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

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fricas [A]  time = 1.08, size = 35, normalized size = 0.92 \[ -\frac {{\left (2 \, b x^{2} + a\right )} \sqrt {b x^{2} + a}}{a^{2} b x^{3} + a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-(2*b*x^2 + a)*sqrt(b*x^2 + a)/(a^2*b*x^3 + a^3*x)

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giac [A]  time = 1.15, size = 50, normalized size = 1.32 \[ -\frac {b x}{\sqrt {b x^{2} + a} a^{2}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 26, normalized size = 0.68 \[ -\frac {2 b \,x^{2}+a}{\sqrt {b \,x^{2}+a}\, a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^2+a)^(3/2),x)

[Out]

-(2*b*x^2+a)/x/(b*x^2+a)^(1/2)/a^2

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maxima [A]  time = 1.29, size = 34, normalized size = 0.89 \[ -\frac {2 \, b x}{\sqrt {b x^{2} + a} a^{2}} - \frac {1}{\sqrt {b x^{2} + a} a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

-2*b*x/(sqrt(b*x^2 + a)*a^2) - 1/(sqrt(b*x^2 + a)*a*x)

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mupad [B]  time = 4.64, size = 35, normalized size = 0.92 \[ -\frac {\sqrt {b\,x^2+a}\,\left (\frac {1}{a}+\frac {2\,b\,x^2}{a^2}\right )}{b\,x^3+a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^2)^(3/2)),x)

[Out]

-((a + b*x^2)^(1/2)*(1/a + (2*b*x^2)/a^2))/(a*x + b*x^3)

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sympy [A]  time = 0.86, size = 46, normalized size = 1.21 \[ - \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**2+a)**(3/2),x)

[Out]

-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) + 1))

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