3.1028 \(\int \frac{1}{x^3 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{4 x^2 \sqrt{c x^4}} \]

[Out]

-1/(4*x^2*Sqrt[c*x^4])

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Rubi [A]  time = 0.0016737, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1, 15, 30} \[ -\frac{1}{4 x^2 \sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*x^2*Sqrt[c*x^4])

Rule 1

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

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^3 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx &=\int \frac{1}{x^3 \sqrt{c x^4}} \, dx\\ &=\frac{x^2 \int \frac{1}{x^5} \, dx}{\sqrt{c x^4}}\\ &=-\frac{1}{4 x^2 \sqrt{c x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0030231, size = 17, normalized size = 1.06 \[ -\frac{c x^2}{4 \left (c x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x^2)/(4*(c*x^4)^(3/2))

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Maple [A]  time = 0.042, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(c*x^4)^(1/2),x)

[Out]

-1/4/x^2/(c*x^4)^(1/2)

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Maxima [A]  time = 0.955371, size = 16, normalized size = 1. \begin{align*} -\frac{1}{4 \, \sqrt{c x^{4}} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4)^(1/2),x, algorithm="maxima")

[Out]

-1/4/(sqrt(c*x^4)*x^2)

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Fricas [A]  time = 1.25511, size = 35, normalized size = 2.19 \begin{align*} -\frac{\sqrt{c x^{4}}}{4 \, c x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(c*x^4)/(c*x^6)

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Sympy [A]  time = 0.546345, size = 19, normalized size = 1.19 \begin{align*} - \frac{1}{4 \sqrt{c} x^{2} \sqrt{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(c*x**4)**(1/2),x)

[Out]

-1/(4*sqrt(c)*x**2*sqrt(x**4))

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Giac [A]  time = 1.19921, size = 11, normalized size = 0.69 \begin{align*} -\frac{1}{4 \, \sqrt{c} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4)^(1/2),x, algorithm="giac")

[Out]

-1/4/(sqrt(c)*x^4)