3.11.36 \(\int \frac {1}{x^2 \sqrt {a+(2+2 c-2 (1+c)) x^4}} \, dx\) [1036]

Optimal. Leaf size=10 \[ -\frac {1}{\sqrt {a} x} \]

[Out]

-1/x/a^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2, 12, 30} \begin {gather*} -\frac {1}{\sqrt {a} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

-(1/(Sqrt[a]*x))

Rule 2

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 \sqrt {a+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {1}{\sqrt {a} x^2} \, dx\\ &=\frac {\int \frac {1}{x^2} \, dx}{\sqrt {a}}\\ &=-\frac {1}{\sqrt {a} x}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{\sqrt {a} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

-(1/(Sqrt[a]*x))

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Maple [A]
time = 0.02, size = 9, normalized size = 0.90

method	result	size
gosper	\(-\frac {1}{x \sqrt {a}}\)	\(9\)
default	\(-\frac {1}{x \sqrt {a}}\)	\(9\)
norman	\(-\frac {1}{x \sqrt {a}}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/a^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/x/a^(1/2)

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Maxima [A]
time = 0.28, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{\sqrt {a} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/a^(1/2),x, algorithm="maxima")

[Out]

-1/(sqrt(a)*x)

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Fricas [A]
time = 0.36, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{\sqrt {a} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/a^(1/2),x, algorithm="fricas")

[Out]

-1/(sqrt(a)*x)

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Sympy [A]
time = 0.01, size = 8, normalized size = 0.80 \begin {gather*} - \frac {1}{\sqrt {a} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/a**(1/2),x)

[Out]

-1/(sqrt(a)*x)

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Giac [A]
time = 3.73, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{\sqrt {a} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/a^(1/2),x, algorithm="giac")

[Out]

-1/(sqrt(a)*x)

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Mupad [B]
time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{\sqrt {a}\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^(1/2)*x^2),x)

[Out]

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

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