3.2932 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^2} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\sqrt{a+b \sqrt{c x^2}}}{x}-\frac{b \sqrt{c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{\sqrt{a} x} \]

[Out]

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

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Rubi [A]  time = 0.0844995, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{\sqrt{a+b \sqrt{c x^2}}}{x}-\frac{b \sqrt{c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{\sqrt{a} x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.52983, size = 56, normalized size = 0.84 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x} - \frac{b \sqrt{c x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**2,x)

[Out]

-sqrt(a + b*sqrt(c*x**2))/x - b*sqrt(c*x**2)*atanh(sqrt(a + b*sqrt(c*x**2))/sqrt
(a))/(sqrt(a)*x)

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Mathematica [A]  time = 0.0339828, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^2,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^2, x]

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Maple [A]  time = 0.016, size = 54, normalized size = 0.8 \[ -{\frac{1}{x} \left ({\it Artanh} \left ({1\sqrt{a+b\sqrt{c{x}^{2}}}{\frac{1}{\sqrt{a}}}} \right ) b\sqrt{c{x}^{2}}+\sqrt{a+b\sqrt{c{x}^{2}}}\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(1/2))^(1/2)/x^2,x)

[Out]

-(arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*b*(c*x^2)^(1/2)+(a+b*(c*x^2)^(1/2))
^(1/2)*a^(1/2))/x/a^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224231, size = 1, normalized size = 0.01 \[ \left [\frac{b x \sqrt{\frac{c}{a}} \log \left (\frac{\sqrt{c x^{2}} b c x + 2 \, a c x - 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} a \sqrt{\frac{c}{a}}}{x^{2}}\right ) - 2 \, \sqrt{\sqrt{c x^{2}} b + a}}{2 \, x}, \frac{b x \sqrt{-\frac{c}{a}} \arctan \left (\frac{a x \sqrt{-\frac{c}{a}}}{\sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a}}\right ) - \sqrt{\sqrt{c x^{2}} b + a}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^2,x, algorithm="fricas")

[Out]

[1/2*(b*x*sqrt(c/a)*log((sqrt(c*x^2)*b*c*x + 2*a*c*x - 2*sqrt(c*x^2)*sqrt(sqrt(c
*x^2)*b + a)*a*sqrt(c/a))/x^2) - 2*sqrt(sqrt(c*x^2)*b + a))/x, (b*x*sqrt(-c/a)*a
rctan(a*x*sqrt(-c/a)/(sqrt(c*x^2)*sqrt(sqrt(c*x^2)*b + a))) - sqrt(sqrt(c*x^2)*b
+ a))/x]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218932, size = 73, normalized size = 1.09 \[ \frac{\frac{b^{2} c \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b \sqrt{c} x + a} b \sqrt{c}}{x}}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^2,x, algorithm="giac")

[Out]

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