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

Optimal. Leaf size=189 \[ -\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c+6 \text {$\#$1}^4 c^2-4 \text {$\#$1}^2 c^3+b+c^4\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}-\text {$\#$1}\right )}{\text {$\#$1} c-\text {$\#$1}^3}\& \right ]-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{c \sqrt {\sqrt {a^2 x^2-b}+a x}} \]

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Rubi [F]  time = 1.47, 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}{x \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 13.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.45, size = 189, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{c \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{c^{3/2}}-\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\text {$\#$1}\right )}{c \text {$\#$1}-\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(c*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (2*ArcTanh[Sqrt[c + Sqrt[a*x
+ Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/c^(3/2) - RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , Lo
g[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] - #1]/(c*#1 - #1^3) & ]

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

[Out]

Timed out

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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/x/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/x/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(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 \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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