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

Optimal. Leaf size=118 \[ \frac {1}{4} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+\frac {b x}{8 \left (\sqrt {a^2 x^4+b}+a x^2\right )^{3/2}}+\frac {5 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a}} \]

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Rubi [F]  time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx &=\int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.28, size = 118, normalized size = 1.00 \begin {gather*} \frac {b x}{8 \left (a x^2+\sqrt {b+a^2 x^4}\right )^{3/2}}+\frac {1}{4} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\frac {5 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x)/(8*(a*x^2 + Sqrt[b + a^2*x^4])^(3/2)) + (x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/4 + (5*Sqrt[b]*ArcTan[(Sqrt[
2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(8*Sqrt[2]*Sqrt[a])

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fricas [A]  time = 4.57, size = 319, normalized size = 2.70 \begin {gather*} \left [\frac {5 \, \sqrt {\frac {1}{2}} b \sqrt {-\frac {b}{a}} \log \left (4 \, a^{2} b x^{4} - 4 \, \sqrt {a^{2} x^{4} + b} a b x^{2} + b^{2} + 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} a^{2} x^{3} \sqrt {-\frac {b}{a}} - \sqrt {\frac {1}{2}} {\left (2 \, a^{3} x^{5} + a b x\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\right ) + 2 \, {\left (2 \, a^{2} x^{5} - 2 \, \sqrt {a^{2} x^{4} + b} a x^{3} + 3 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, b}, -\frac {5 \, \sqrt {\frac {1}{2}} b \sqrt {\frac {b}{a}} \arctan \left (-\frac {{\left (\sqrt {\frac {1}{2}} a x^{2} \sqrt {\frac {b}{a}} - \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} \sqrt {\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{b x}\right ) - {\left (2 \, a^{2} x^{5} - 2 \, \sqrt {a^{2} x^{4} + b} a x^{3} + 3 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{8 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(5*sqrt(1/2)*b*sqrt(-b/a)*log(4*a^2*b*x^4 - 4*sqrt(a^2*x^4 + b)*a*b*x^2 + b^2 + 4*(2*sqrt(1/2)*sqrt(a^2*
x^4 + b)*a^2*x^3*sqrt(-b/a) - sqrt(1/2)*(2*a^3*x^5 + a*b*x)*sqrt(-b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))) + 2*(
2*a^2*x^5 - 2*sqrt(a^2*x^4 + b)*a*x^3 + 3*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/b, -1/8*(5*sqrt(1/2)*b*sqrt(b/
a)*arctan(-(sqrt(1/2)*a*x^2*sqrt(b/a) - sqrt(1/2)*sqrt(a^2*x^4 + b)*sqrt(b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))
/(b*x)) - (2*a^2*x^5 - 2*sqrt(a^2*x^4 + b)*a*x^3 + 3*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/b]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a^2*x^4 + b)/sqrt(a*x^2 + 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 {a^2\,x^4+b}}{\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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