3.2927 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.124414, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^2}}}{4 a \sqrt{c x^2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^2]]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5917, size = 83, normalized size = 0.86 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{2 x^{2}} - \frac{b c \sqrt{a + b \sqrt{c x^{2}}}}{4 a \sqrt{c x^{2}}} + \frac{b^{2} c \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**3,x)

[Out]

-sqrt(a + b*sqrt(c*x**2))/(2*x**2) - b*c*sqrt(a + b*sqrt(c*x**2))/(4*a*sqrt(c*x*
*2)) + b**2*c*atanh(sqrt(a + b*sqrt(c*x**2))/sqrt(a))/(4*a**(3/2))

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Mathematica [A]  time = 0.0317273, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^3,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^3, x]

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Maple [A]  time = 0.008, size = 72, normalized size = 0.7 \[ -{\frac{1}{4\,{x}^{2}} \left ( -{\it Artanh} \left ({1\sqrt{a+b\sqrt{c{x}^{2}}}{\frac{1}{\sqrt{a}}}} \right ) c{x}^{2}a{b}^{2}+{a}^{{\frac{3}{2}}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}}+{a}^{{\frac{5}{2}}}\sqrt{a+b\sqrt{c{x}^{2}}} \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(1/2))^(1/2)/x^3,x)

[Out]

-1/4*(-arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*c*x^2*a*b^2+a^(3/2)*(a+b*(c*x^
2)^(1/2))^(3/2)+a^(5/2)*(a+b*(c*x^2)^(1/2))^(1/2))/x^2/a^(5/2)

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223849, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} b^{2} c x^{2} \log \left (\frac{\sqrt{c x^{2}} \sqrt{a} b + 2 \, \sqrt{\sqrt{c x^{2}} b + a} a + 2 \, a^{\frac{3}{2}}}{x}\right ) - 2 \,{\left (\sqrt{c x^{2}} a b + 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{8 \, a^{2} x^{2}}, \frac{\sqrt{-a} b^{2} c x^{2} \arctan \left (\frac{a}{\sqrt{\sqrt{c x^{2}} b + a} \sqrt{-a}}\right ) -{\left (\sqrt{c x^{2}} a b + 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{4 \, a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^3,x, algorithm="fricas")

[Out]

[1/8*(sqrt(a)*b^2*c*x^2*log((sqrt(c*x^2)*sqrt(a)*b + 2*sqrt(sqrt(c*x^2)*b + a)*a
+ 2*a^(3/2))/x) - 2*(sqrt(c*x^2)*a*b + 2*a^2)*sqrt(sqrt(c*x^2)*b + a))/(a^2*x^2
), 1/4*(sqrt(-a)*b^2*c*x^2*arctan(a/(sqrt(sqrt(c*x^2)*b + a)*sqrt(-a))) - (sqrt(
c*x^2)*a*b + 2*a^2)*sqrt(sqrt(c*x^2)*b + a))/(a^2*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**3, x)

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GIAC/XCAS [A]  time = 0.218893, size = 122, normalized size = 1.26 \[ -\frac{\frac{b^{3} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} + \sqrt{b \sqrt{c} x + a} a b^{3} c^{\frac{3}{2}}}{a b^{2} c x^{2}}}{4 \, b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^3,x, algorithm="giac")

[Out]

-1/4*(b^3*c^(3/2)*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*sqrt
(c)*x + a)^(3/2)*b^3*c^(3/2) + sqrt(b*sqrt(c)*x + a)*a*b^3*c^(3/2))/(a*b^2*c*x^2
))/(b*sqrt(c))