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

Optimal. Leaf size=269 \[ \frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} (6 b-16 a x)+\sqrt {a^2 x^2-b} \left (8 \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}-16 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+(8 a x-9 b) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{12 a \sqrt {a^2 x^2-b}+12 a^2 x}+\frac {3 b \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{4 a} \]

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Rubi [F]  time = 0.08, 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 {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.23, size = 188, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-\frac {9 b}{\sqrt {\sqrt {a^2 x^2-b}+a x}}+\frac {6 b}{\sqrt {a^2 x^2-b}+a x}+8 \sqrt {\sqrt {a^2 x^2-b}+a x}-16\right )-9 b \log \left (1-\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}\right )+9 b \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}+1\right )}{24 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(-16 + (6*b)/(a*x + Sqrt[-b + a^2*x^2]) - (9*b)/Sqrt[a*x + Sqrt[-b
+ a^2*x^2]] + 8*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - 9*b*Log[1 - 1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]] + 9
*b*Log[1 + 1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(24*a)

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IntegrateAlgebraic [A]  time = 0.41, size = 269, normalized size = 1.00 \begin {gather*} \frac {(6 b-16 a x) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+(-9 b+8 a x) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (-16 \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+8 \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{12 a^2 x+12 a \sqrt {-b+a^2 x^2}}+\frac {3 b \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

((6*b - 16*a*x)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-9*b + 8*a*x)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[
1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + Sqrt[-b + a^2*x^2]*(-16*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + 8*Sqr
t[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(12*a^2*x + 12*a*Sqrt[-b + a^2*x^2]) +
(3*b*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(4*a)

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fricas [A]  time = 0.51, size = 152, normalized size = 0.57 \begin {gather*} \frac {9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (6 \, a x - {\left (9 \, a x - 9 \, \sqrt {a^{2} x^{2} - b} - 8\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 6 \, \sqrt {a^{2} x^{2} - b} - 16\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{24 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*b*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) + 1) - 9*b*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) -
1) + 2*(6*a*x - (9*a*x - 9*sqrt(a^2*x^2 - b) - 8)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 6*sqrt(a^2*x^2 - b) - 16)*s
qrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1))/a

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

[Out]

Timed out

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {1+\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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