3.29.50 \(\int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\)

Optimal. Leaf size=293 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{4 a c^{3/2}}+\frac {\left (48 a^2 c x^2-16 a c^3 x-6 b c\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+\sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c} \left (8 a c^2 x-15 b\right )+\sqrt {a^2 x^2+b} \left (\left (48 a c x-16 c^3\right ) \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}+8 c^2 \sqrt {\sqrt {a^2 x^2+b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}\right )}{60 a c \sqrt {a^2 x^2+b}+60 a^2 c x} \]

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Rubi [F]  time = 0.07, 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 \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.59, size = 220, normalized size = 0.75 \begin {gather*} -\frac {2 \left (-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{8 c^{3/2}}-\frac {1}{5} \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right )^{5/2}+\frac {1}{3} c \left (\sqrt {\sqrt {a^2 x^2+b}+a x}+c\right )^{3/2}+\frac {b \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{8 c \sqrt {\sqrt {a^2 x^2+b}+a x}}+\frac {b \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{4 \left (\sqrt {a^2 x^2+b}+a x\right )}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((b*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(4*(a*x + Sqrt[b + a^2*x^2])) + (b*Sqrt[c + Sqrt[a*x + Sqrt[b
+ a^2*x^2]]])/(8*c*Sqrt[a*x + Sqrt[b + a^2*x^2]]) + (c*(c + Sqrt[a*x + Sqrt[b + a^2*x^2]])^(3/2))/3 - (c + Sq
rt[a*x + Sqrt[b + a^2*x^2]])^(5/2)/5 - (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(8*c^(3/2)
)))/a

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IntegrateAlgebraic [A]  time = 0.49, size = 293, normalized size = 1.00 \begin {gather*} \frac {\left (-6 b c-16 a c^3 x+48 a^2 c x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-15 b+8 a c^2 x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (-16 c^3+48 a c x\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+8 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{60 a^2 c x+60 a c \sqrt {b+a^2 x^2}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{4 a c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*b*c - 16*a*c^3*x + 48*a^2*c*x^2)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-15*b + 8*a*c^2*x)*Sqrt[a*x +
Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqrt[b + a^2*x^2]*((-16*c^3 + 48*a*c*x)*Sqrt[c +
Sqrt[a*x + Sqrt[b + a^2*x^2]]] + 8*c^2*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])
)/(60*a^2*c*x + 60*a*c*Sqrt[b + a^2*x^2]) + (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(4*a*
c^(3/2))

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fricas [A]  time = 0.58, size = 359, normalized size = 1.23 \begin {gather*} \left [\frac {15 \, b \sqrt {c} \log \left (-2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) - 2 \, {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{120 \, a c^{2}}, -\frac {15 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{60 \, a c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/120*(15*b*sqrt(c)*log(-2*(a*sqrt(c)*x - sqrt(a^2*x^2 + b)*sqrt(c))*sqrt(a*x + sqrt(a^2*x^2 + b))*sqrt(c + s
qrt(a*x + sqrt(a^2*x^2 + b))) - 2*(a*c*x - sqrt(a^2*x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2 + b)) + b) - 2*(16*c^4
- 54*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 + 15*a*c*x - 15*sqrt(a^2*x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2
+ b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^2), -1/60*(15*b*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*
x + sqrt(a^2*x^2 + b)))/c) + (16*c^4 - 54*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 + 15*a*c*x - 15*sqrt(a^2*
x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c+(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 \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b))), x)

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