3.1037 \(\int \frac{1}{x^3 \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{2 \sqrt{a} x^2} \]

[Out]

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

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Rubi [A]  time = 0.001611, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ -\frac{1}{2 \sqrt{a} x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

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

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^3 \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac{1}{\sqrt{a} x^3} \, dx\\ &=\frac{\int \frac{1}{x^3} \, dx}{\sqrt{a}}\\ &=-\frac{1}{2 \sqrt{a} x^2}\\ \end{align*}

Mathematica [A]  time = 0.0004923, size = 12, normalized size = 1. \[ -\frac{1}{2 \sqrt{a} x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

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

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Maple [A]  time = 0.04, size = 9, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/a^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.946884, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2 \, \sqrt{a} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28153, size = 27, normalized size = 2.25 \begin{align*} -\frac{1}{2 \, \sqrt{a} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.063928, size = 12, normalized size = 1. \begin{align*} - \frac{1}{2 \sqrt{a} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20006, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2 \, \sqrt{a} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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