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

Optimal. Leaf size=81 \[ \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\sqrt {2} \sqrt {a}} \]

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Rubi [A]  time = 0.12, antiderivative size = 47, normalized size of antiderivative = 0.58, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2132, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 47, normalized size = 0.58 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {\sqrt {a^2 x^4+b}+a x^2}}\right )}{\sqrt {2} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.39, size = 81, normalized size = 1.00 \begin {gather*} \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt
[a])

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fricas [A]  time = 1.98, size = 135, normalized size = 1.67 \begin {gather*} \left [\frac {\sqrt {2} \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{4 \, \sqrt {a}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(2)*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 + 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*sqrt(a^2*x^4 + b)*sq
rt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b)/sqrt(a), -1/2*sqrt(2)*sqrt(-1/a)*arctan(1/2*sqrt(2)*sqrt(a*x^2 +
sqrt(a^2*x^4 + b))*sqrt(-1/a)/x)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/sqrt(a^2*x^4 + b), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/sqrt(a^2*x^4 + b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\sqrt {a^2\,x^4+b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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