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

Optimal. Leaf size=163 \[ \frac {1}{2} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (3-2 a x^2\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{4 b}+\frac {5 \tanh ^{-1}\left (\sqrt {2} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{4 \sqrt {2} b} \]

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Rubi [F]  time = 0.54, 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 {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 21.92, size = 11560, normalized size = 70.92 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 3.13, size = 163, normalized size = 1.00 \begin {gather*} \frac {\left (3-2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{4 b}+\frac {1}{2} x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+\frac {5 \tanh ^{-1}\left (\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{4 \sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 - 2*a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(4*b) + (x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*Sqr
t[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/2 + (5*ArcTanh[Sqrt[2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^
2)/b^2]]])/(4*Sqrt[2]*b)

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fricas [A]  time = 15.21, size = 168, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (2 \, a x^{2} - 2 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 3\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} - 5 \, \sqrt {2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right )}{16 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*(2*a*x^2 - 2*b*x*sqrt((a^2*x^2 - a)/b^2) - 3)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) - 5*sqrt(2)*l
og(4*a*x^2 - 4*b*x*sqrt((a^2*x^2 - a)/b^2) - 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(2*sqrt(2)*b*x*sqrt((
a^2*x^2 - a)/b^2) - sqrt(2)*(2*a*x^2 - 1)) - 1))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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