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

Optimal. Leaf size=53 \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(b*x^2+a)*arctan(x*b^(1/2)/a^(1/2))/a^(1/2)/b^(1/2)/((b*x^2+a)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1088, 205} \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

((a + b*x^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1088

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^p/(b + 2*c*x^2)^(2*p), In
t[(b + 2*c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 44, normalized size = 0.83 \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 67, normalized size = 1.26 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a))/(a*b), sqrt(a*b)*arctan(sqrt(a*b)*x/a)/(a*b)]

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giac [A]  time = 0.18, size = 23, normalized size = 0.43 \[ \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(b*x/sqrt(a*b))*sgn(b*x^2 + a)/sqrt(a*b)

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maple [A]  time = 0.00, size = 34, normalized size = 0.64 \[ \frac {\left (b \,x^{2}+a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((b*x^2+a)^2)^(1/2)*(b*x^2+a)/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.03, size = 15, normalized size = 0.28 \[ \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(b*x/sqrt(a*b))/sqrt(a*b)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)^2)^(1/2),x)

[Out]

int(1/((a + b*x^2)^2)^(1/2), x)

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sympy [A]  time = 0.18, size = 53, normalized size = 1.00 \[ - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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