3.1029 \(\int \frac{1}{x^4 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{5 x^3 \sqrt{c x^4}} \]

[Out]

-1/(5*x^3*Sqrt[c*x^4])

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Rubi [A]  time = 0.0016077, 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} \[ -\frac{1}{5 x^3 \sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(5*x^3*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^4 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx &=\int \frac{1}{x^4 \sqrt{c x^4}} \, dx\\ &=\frac{x^2 \int \frac{1}{x^6} \, dx}{\sqrt{c x^4}}\\ &=-\frac{1}{5 x^3 \sqrt{c x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0018054, size = 15, normalized size = 0.94 \[ -\frac{c x}{5 \left (c x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x)/(5*(c*x^4)^(3/2))

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Maple [A]  time = 0.041, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{5\,{x}^{3}}{\frac{1}{\sqrt{c{x}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/x^3/(c*x^4)^(1/2)

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Maxima [A]  time = 0.943107, size = 16, normalized size = 1. \begin{align*} -\frac{1}{5 \, \sqrt{c x^{4}} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/(sqrt(c*x^4)*x^3)

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Fricas [A]  time = 1.31814, size = 35, normalized size = 2.19 \begin{align*} -\frac{\sqrt{c x^{4}}}{5 \, c x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(c*x^4)/(c*x^7)

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Sympy [A]  time = 0.623809, size = 19, normalized size = 1.19 \begin{align*} - \frac{1}{5 \sqrt{c} x^{3} \sqrt{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(5*sqrt(c)*x**3*sqrt(x**4))

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Giac [A]  time = 1.12828, size = 11, normalized size = 0.69 \begin{align*} -\frac{1}{5 \, \sqrt{c} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/(sqrt(c)*x^5)