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]

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

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Rubi [A]  time = 0.0104717, antiderivative size = 44, normalized size of antiderivative = 1., 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 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]

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^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*}

Mathematica [A]  time = 0.0063714, 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]

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

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Maple [A]  time = 0.044, size = 26, normalized size = 0.6 \begin{align*} -{\frac{-2\,b{x}^{2}+a}{3\,{a}^{2}{x}^{3}}\sqrt{b{x}^{2}+a}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.49855, size = 61, normalized size = 1.39 \begin{align*} \frac{{\left (2 \, b x^{2} - a\right )} \sqrt{b x^{2} + a}}{3 \, a^{2} x^{3}} \end{align*}

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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Sympy [A]  time = 0.753629, size = 46, normalized size = 1.05 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.19944, size = 74, normalized size = 1.68 \begin{align*} \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}} \end{align*}

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