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

Optimal. Leaf size=77 \[ \frac {4 \sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{a}-\frac {4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )}{a} \]

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Rubi [F]  time = 0.33, 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 {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\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]]]/Sqrt[b + a^2*x^2],x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 74, normalized size = 0.96 \begin {gather*} \frac {4 \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}}{\sqrt {c}}\right )\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.28, size = 77, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a}-\frac {4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/a - (4*Sqrt[c]*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqr
t[c]])/a

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fricas [A]  time = 0.65, size = 202, normalized size = 2.62 \begin {gather*} \left [\frac {2 \, {\left (\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 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right )}}{a}, \frac {4 \, {\left (\sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right )}}{a}\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)/(a^2*x^2+b)^(1/2),x, algorithm="fricas")

[Out]

[2*(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 + sqrt(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*sqrt(c + sqrt(a*x
+ sqrt(a^2*x^2 + b))))/a, 4*(sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/c) + sqrt(c + s
qrt(a*x + sqrt(a^2*x^2 + b))))/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((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\sqrt {a^{2} x^{2}+b}}\, dx\]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{\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)/(a^2*x^2+b)^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

int((c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2)/(b + a^2*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 {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{\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)/(a**2*x**2+b)**(1/2),x)

[Out]

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

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