3.1.85 \(\int \frac {1}{(a+b x^2)^{3/2}} \, dx\) [85]

Optimal. Leaf size=16 \[ \frac {x}{a \sqrt {a+b x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \begin {gather*} \frac {x}{a \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

Rule 197

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x}{a \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

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Maple [A]
time = 0.05, size = 15, normalized size = 0.94

method	result	size
gosper	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)
default	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)
trager	\(\frac {x}{a \sqrt {b \,x^{2}+a}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {b x^{2} + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(b*x^2 + a)*a)

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Fricas [A]
time = 0.55, size = 23, normalized size = 1.44 \begin {gather*} \frac {\sqrt {b x^{2} + a} x}{a b x^{2} + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)*x/(a*b*x^2 + a^2)

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Sympy [A]
time = 0.31, size = 17, normalized size = 1.06 \begin {gather*} \frac {x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(a**(3/2)*sqrt(1 + b*x**2/a))

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Giac [A]
time = 0.68, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {b x^{2} + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(b*x^2 + a)*a)

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Mupad [B]
time = 0.04, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{a\,\sqrt {b\,x^2+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^2)^(3/2),x)

[Out]

x/(a*(a + b*x^2)^(1/2))

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