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

Optimal. Leaf size=71 \[ -\frac {\sqrt {2} \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt {a} \sqrt {x \sqrt {a^2 x^2-b}+a x^2}+a x\right )}{\sqrt {a}} \]

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Rubi [F]  time = 1.19, 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 {a x^2+x \sqrt {-b+a^2 x^2}}}{x \sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 104, normalized size = 1.46 \begin {gather*} \frac {\sqrt {2} \sqrt {x \left (\sqrt {a^2 x^2-b}+a x\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}}\right )}{\sqrt {a} \sqrt {x} \sqrt {\sqrt {a^2 x^2-b}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.56, size = 71, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \log \left (a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt {a} \sqrt {a x^2+x \sqrt {-b+a^2 x^2}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 18.60, size = 142, normalized size = 2.00 \begin {gather*} \left [\frac {\sqrt {2} \log \left (-4 \, a^{2} x^{2} - 4 \, \sqrt {a^{2} x^{2} - b} a x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x} + b\right )}{2 \, \sqrt {a}}, -\sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b}}}{x \sqrt {a^{2} x^{2}-b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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