3.1020 \(\int \frac {1}{x^4 \sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=44 \[ \frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3} \]

[Out]

-1/3*(b*x^2+a)^(1/2)/a/x^3+2/3*b*(b*x^2+a)^(1/2)/a^2/x

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5, 271, 264} \[ \frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[a + b*x^2]/(3*a*x^3) + (2*b*Sqrt[a + b*x^2])/(3*a^2*x)

Rule 5

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

Rule 271

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.66 \[ -\frac {\left (a-2 b x^2\right ) \sqrt {a+b x^2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 27, normalized size = 0.61 \[ \frac {{\left (2 \, b x^{2} - a\right )} \sqrt {b x^{2} + a}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*b*x^2 - a)*sqrt(b*x^2 + a)/(a^2*x^3)

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giac [A]  time = 0.28, size = 55, normalized size = 1.25 \[ \frac {4 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*b^(3/2)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3

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maple [A]  time = 0.00, size = 26, normalized size = 0.59 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b \,x^{2}+a \right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x^2+a)^(1/2)*(-2*b*x^2+a)/a^2/x^3

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maxima [A]  time = 1.00, size = 36, normalized size = 0.82 \[ \frac {2 \, \sqrt {b x^{2} + a} b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x^2 + a)*b/(a^2*x) - 1/3*sqrt(b*x^2 + a)/(a*x^3)

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mupad [B]  time = 4.55, size = 25, normalized size = 0.57 \[ -\frac {\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^(1/2)),x)

[Out]

-((a + b*x^2)^(1/2)*(a - 2*b*x^2))/(3*a^2*x^3)

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sympy [A]  time = 1.05, size = 46, normalized size = 1.05 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*a*x**2) + 2*b**(3/2)*sqrt(a/(b*x**2) + 1)/(3*a**2)

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