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

Optimal. Leaf size=140 \[ 2 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^5 a^2-2 \text {$\#$1}^4 b+b^2\& ,\frac {\text {$\#$1} a^2 \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )+b \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )}{-2 \text {$\#$1}^4+5 \text {$\#$1} a^2+2 b}\& \right ]-a \log \left (\sqrt {a^2 x^2+b}-a x\right ) \]

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Rubi [F]  time = 4.84, 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^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

b*Defer[Int][x^2/(b + 2*a*x^5 - x^8), x] + a^2*Defer[Int][x^4/(b + 2*a*x^5 - x^8), x] + a*Defer[Int][(x^3*Sqrt
[b + a^2*x^2])/(b + 2*a*x^5 - x^8), x] + Defer[Int][(x^6*Sqrt[b + a^2*x^2])/(-b - 2*a*x^5 + x^8), x] + b*Defer
[Int][Sqrt[a*x - Sqrt[b + a^2*x^2]]/(b + 2*a*x^5 - x^8), x] + a^2*Defer[Int][(x^2*Sqrt[a*x - Sqrt[b + a^2*x^2]
])/(b + 2*a*x^5 - x^8), x] + a*Defer[Int][(x*Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(b + 2*a*x^5 - x
^8), x] - Defer[Int][(x^4*Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(b + 2*a*x^5 - x^8), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx &=\int \left (\frac {x^2 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8}+\frac {x^3 \sqrt {b+a^2 x^2} \left (-a+x^3\right )}{-b-2 a x^5+x^8}+\frac {a x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}-\frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {\left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {x^2 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8} \, dx+\int \frac {x^3 \sqrt {b+a^2 x^2} \left (-a+x^3\right )}{-b-2 a x^5+x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {\left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\\ &=a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \left (\frac {b x^2}{b+2 a x^5-x^8}+\frac {a^2 x^4}{b+2 a x^5-x^8}\right ) \, dx+\int \left (\frac {a x^3 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8}+\frac {x^6 \sqrt {b+a^2 x^2}}{-b-2 a x^5+x^8}\right ) \, dx+\int \left (\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {a^2 x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=a \int \frac {x^3 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx+a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^4}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+b \int \frac {x^2}{b+2 a x^5-x^8} \, dx+b \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {x^6 \sqrt {b+a^2 x^2}}{-b-2 a x^5+x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 1.61, size = 140, normalized size = 1.00 \begin {gather*} -a \log \left (-a x+\sqrt {b+a^2 x^2}\right )+2 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Log[-(a*x) + Sqrt[b + a^2*x^2]]) + 2*a*RootSum[b^2 - 2*b*#1^4 - 4*a^2*#1^5 + #1^8 & , (b*Log[Sqrt[a*x - Sq
rt[b + a^2*x^2]] - #1] + a^2*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1)/(2*b + 5*a^2*#1 - 2*#1^4) & ]

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

[Out]

Timed out

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

[Out]

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

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}+b}}{x^{2}-\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+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^{2} + b}}{x^{2} - \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((a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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