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

Optimal. Leaf size=21 \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

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Rubi [A]  time = 0.0042075, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 264} \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^4]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

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

Mathematica [A]  time = 0.0038213, size = 21, normalized size = 1. \[ x \sqrt{x^2+1} \left (-\sqrt{\frac{a}{x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^4]/Sqrt[1 + x^2],x]

[Out]

-(Sqrt[a/x^4]*x*Sqrt[1 + x^2])

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -x\sqrt{{\frac{a}{{x}^{4}}}}\sqrt{{x}^{2}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.5862, size = 31, normalized size = 1.48 \begin{align*} -\frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.982961, size = 65, normalized size = 3.1 \begin{align*} -x^{2} \sqrt{\frac{a}{x^{4}}} - \sqrt{x^{2} + 1} x \sqrt{\frac{a}{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x**4)/sqrt(x**2 + 1), x)

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Giac [A]  time = 1.15102, size = 30, normalized size = 1.43 \begin{align*} \frac{2 \, \sqrt{a}}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)/((x - sqrt(x^2 + 1))^2 - 1)