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

Optimal. Leaf size=196 \[ \frac {\left (-a x^2-2\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{b x}+\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-\frac {\sqrt {a} \log \left (b \left (-\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}\right )+\sqrt {2} \sqrt {a} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-a x\right )}{\sqrt {2} b} \]

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Rubi [F]  time = 1.71, 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}}}{x \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]/(x*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]/(x*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}}}{x \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}}}{x \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 [A]  time = 11.66, size = 384, normalized size = 1.96 \begin {gather*} \frac {\sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (-8 a \left (10 a^2 x^4+2 a x^2 \left (5 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-6\right )-7 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2\right )-5 \sqrt {2} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (2 a^2 x^3+2 a x \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-1\right )-b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt {2} a x}\right )\right )+9 \sqrt {2} a^{3/2} x \left (4 a^2 x^4+a x^2 \left (4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-5\right )-3 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+1\right ) \sinh ^{-1}\left (\frac {b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x}{\sqrt {a}}\right )}{8 b x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Sqrt[2]*a^(3/2)*x*(1 + 4*a^2*x^4 - 3*b*x*Sqrt[(a*(-1 + a*x^2))/b^2] + a*x^2*(-5 + 4*b*x*Sqrt[(a*(-1 + a*x^2
))/b^2]))*ArcSinh[(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])/Sqrt[a]] + Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]
*(-8*a*(2 + 10*a^2*x^4 - 7*b*x*Sqrt[(a*(-1 + a*x^2))/b^2] + 2*a*x^2*(-6 + 5*b*x*Sqrt[(a*(-1 + a*x^2))/b^2])) -
5*Sqrt[2]*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(2*a^2*x^3 - b*Sqrt[(a*(-1 + a*x^2))/b^2] + 2*a*x*(-
1 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))*ArcTanh[Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)]))/(
8*b*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))

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IntegrateAlgebraic [A]  time = 3.23, size = 196, normalized size = 1.00 \begin {gather*} \frac {\left (-2-a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{b 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 {\sqrt {a} \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 - a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(b*x) + 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]] - (Sqrt[a]*Log[-(a*x) - b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2] + Sqrt[2]*S
qrt[a]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(Sqrt[2]*b)

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fricas [A]  time = 26.49, size = 295, normalized size = 1.51 \begin {gather*} \left [\frac {\sqrt {2} \sqrt {a} x \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{4 \, b x}, -\frac {\sqrt {2} \sqrt {-a} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{2 \, b x}\right ] \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)/x/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(2)*sqrt(a)*x*log(-4*a^2*x^2 - 4*a*b*x*sqrt((a^2*x^2 - a)/b^2) - 2*(sqrt(2)*a^(3/2)*x + sqrt(2)*sqrt
(a)*b*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) + a) - 4*sqrt(a*x^2 + b*x*sqrt((a^2*x
^2 - a)/b^2))*(a*x^2 - b*x*sqrt((a^2*x^2 - a)/b^2) + 2))/(b*x), -1/2*(sqrt(2)*sqrt(-a)*x*arctan(1/2*sqrt(2)*sq
rt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*sqrt(-a)/(a*x)) + 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(a*x^2 -
b*x*sqrt((a^2*x^2 - a)/b^2) + 2))/(b*x)]

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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} 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)/x/(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), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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