3.4.84 \(\int \frac {\sqrt {\frac {a}{x^4}}}{\sqrt {1+x^2}} \, dx\) [384]

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

[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, 270} \begin {gather*} x \sqrt {x^2+1} \left (-\sqrt {\frac {a}{x^4}}\right ) \end {gather*}

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 270

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 = 0.03, size = 21, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {a}{x^4}} x \sqrt {1+x^2} \end {gather*}

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 = 0.24, size = 18, normalized size = 0.86

method	result	size
gosper	\(-x \sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{2}+1}\)	\(18\)
default	\(-x \sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{2}+1}\)	\(18\)
meijerg	\(-x \sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{2}+1}\)	\(18\)
risch	\(-x \sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{2}+1}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 23, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {a} x^{2} + \sqrt {a}}{\sqrt {x^{2} + 1} x} \end {gather*}

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 = 0.33, size = 30, normalized size = 1.43 \begin {gather*} -x^{2} \sqrt {\frac {a}{x^{4}}} - \sqrt {x^{2} + 1} x \sqrt {\frac {a}{x^{4}}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {x^{2} + 1}}\, dx \end {gather*}

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 = 3.55, size = 22, normalized size = 1.05 \begin {gather*} \frac {2 \, \sqrt {a}}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} \end {gather*}

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

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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