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

Optimal. Leaf size=21 \[ -\frac {\sqrt {a+c x^4}}{2 a x^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4, 264} \begin {gather*} -\frac {\sqrt {a+c x^4}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + c*x^4]/(2*a*x^2)

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+c x^4}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Sqrt[a + c*x^4]/(a*x^2)

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IntegrateAlgebraic [A]  time = 0.06, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+c x^4}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Sqrt[a + c*x^4]/(a*x^2)

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fricas [A]  time = 2.55, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^4 + a)/(a*x^2)

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giac [A]  time = 0.17, size = 31, normalized size = 1.48 \begin {gather*} \frac {\sqrt {c}}{{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {c \,x^{4}+a}}{2 a \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c x^{4} + a}}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^4 + a)/(a*x^2)

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mupad [B]  time = 4.51, size = 17, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c\,x^4+a}}{2\,a\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + c*x^4)^(1/2)),x)

[Out]

-(a + c*x^4)^(1/2)/(2*a*x^2)

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sympy [A]  time = 0.84, size = 20, normalized size = 0.95 \begin {gather*} - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(2*a)

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