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]

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

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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 = 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*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ x \sqrt {x^2+1} \left (-\sqrt {\frac {a}{x^4}}\right ) \]

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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fricas [A]  time = 0.43, size = 30, normalized size = 1.43 \[ -x^{2} \sqrt {\frac {a}{x^{4}}} - \sqrt {x^{2} + 1} x \sqrt {\frac {a}{x^{4}}} \]

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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giac [A]  time = 0.18, size = 22, normalized size = 1.05 \[ \frac {2 \, \sqrt {a}}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} \]

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)

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ -\sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{2}+1}\, x \]

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 = 1.97, size = 23, normalized size = 1.10 \[ -\frac {\sqrt {a} x^{2} + \sqrt {a}}{\sqrt {x^{2} + 1} x} \]

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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mupad [B]  time = 2.87, size = 18, normalized size = 0.86 \[ -\sqrt {a}\,x\,\sqrt {x^2+1}\,\sqrt {\frac {1}{x^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^4)^(1/2)/(x^2 + 1)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {x^{2} + 1}}\, dx \]

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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