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

Optimal. Leaf size=43 \[ \frac {2 b \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{a x} \]

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Rubi [F]  time = 1.77, 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 {-\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[1/(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]]),x]

[Out]

Defer[Int][1/(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]]), x]

Rubi steps

\begin {align*} \int \frac {1}{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}}}} \, dx &=\int \frac {1}{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}}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.65, size = 72, normalized size = 1.67 \begin {gather*} \frac {2 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right )}{\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(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]]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.89, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(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]]),x]

[Out]

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

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fricas [A]  time = 1.17, size = 37, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} b}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}\,\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(1/(x*(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/2)*((a^2*x^2)/b^2 - a/b^2)^(1/2)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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