3.2933 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^4} \, dx\)

Optimal. Leaf size=144 \[ -\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(3*x^3) + (b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(8*a^2*x) - (
b*(c*x^2)^(3/2)*Sqrt[a + b*Sqrt[c*x^2]])/(12*a*c*x^5) - (b^3*(c*x^2)^(3/2)*ArcTa
nh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(8*a^(5/2)*x^3)

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Rubi [A]  time = 0.164832, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^2]]/x^4,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(3*x^3) + (b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(8*a^2*x) - (
b*(c*x^2)^(3/2)*Sqrt[a + b*Sqrt[c*x^2]])/(12*a*c*x^5) - (b^3*(c*x^2)^(3/2)*ArcTa
nh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])/(8*a^(5/2)*x^3)

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Rubi in Sympy [A]  time = 15.6864, size = 124, normalized size = 0.86 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{3 x^{3}} - \frac{b \left (c x^{2}\right )^{\frac{3}{2}} \sqrt{a + b \sqrt{c x^{2}}}}{12 a c x^{5}} + \frac{b^{2} c \sqrt{a + b \sqrt{c x^{2}}}}{8 a^{2} x} - \frac{b^{3} \left (c x^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**4,x)

[Out]

-sqrt(a + b*sqrt(c*x**2))/(3*x**3) - b*(c*x**2)**(3/2)*sqrt(a + b*sqrt(c*x**2))/
(12*a*c*x**5) + b**2*c*sqrt(a + b*sqrt(c*x**2))/(8*a**2*x) - b**3*(c*x**2)**(3/2
)*atanh(sqrt(a + b*sqrt(c*x**2))/sqrt(a))/(8*a**(5/2)*x**3)

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Mathematica [A]  time = 0.0324504, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^4,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^4, x]

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Maple [A]  time = 0.019, size = 97, normalized size = 0.7 \[ -{\frac{1}{24\,{x}^{3}} \left ( 3\,{a}^{9/2}\sqrt{a+b\sqrt{c{x}^{2}}}+8\,{a}^{7/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}-3\,{a}^{5/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}+3\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{2}{b}^{3} \left ( c{x}^{2} \right ) ^{3/2} \right ){a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(1/2))^(1/2)/x^4,x)

[Out]

-1/24*(3*a^(9/2)*(a+b*(c*x^2)^(1/2))^(1/2)+8*a^(7/2)*(a+b*(c*x^2)^(1/2))^(3/2)-3
*a^(5/2)*(a+b*(c*x^2)^(1/2))^(5/2)+3*arctanh((a+b*(c*x^2)^(1/2))^(1/2)/a^(1/2))*
a^2*b^3*(c*x^2)^(3/2))/x^3/a^(9/2)

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224765, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} c x^{3} \sqrt{\frac{c}{a}} \log \left (\frac{\sqrt{c x^{2}} b c x + 2 \, a c x - 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} a \sqrt{\frac{c}{a}}}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{48 \, a^{2} x^{3}}, \frac{3 \, b^{3} c x^{3} \sqrt{-\frac{c}{a}} \arctan \left (\frac{a x \sqrt{-\frac{c}{a}}}{\sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a}}\right ) +{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{24 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^4,x, algorithm="fricas")

[Out]

[1/48*(3*b^3*c*x^3*sqrt(c/a)*log((sqrt(c*x^2)*b*c*x + 2*a*c*x - 2*sqrt(c*x^2)*sq
rt(sqrt(c*x^2)*b + a)*a*sqrt(c/a))/x^2) + 2*(3*b^2*c*x^2 - 2*sqrt(c*x^2)*a*b - 8
*a^2)*sqrt(sqrt(c*x^2)*b + a))/(a^2*x^3), 1/24*(3*b^3*c*x^3*sqrt(-c/a)*arctan(a*
x*sqrt(-c/a)/(sqrt(c*x^2)*sqrt(sqrt(c*x^2)*b + a))) + (3*b^2*c*x^2 - 2*sqrt(c*x^
2)*a*b - 8*a^2)*sqrt(sqrt(c*x^2)*b + a))/(a^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**4,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**4, x)

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GIAC/XCAS [A]  time = 0.220577, size = 154, normalized size = 1.07 \[ \frac{\frac{3 \, b^{4} c^{2} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} b^{4} c^{2} - 8 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a b^{4} c^{2} - 3 \, \sqrt{b \sqrt{c} x + a} a^{2} b^{4} c^{2}}{a^{2} b^{3} c^{\frac{3}{2}} x^{3}}}{24 \, b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^4,x, algorithm="giac")

[Out]

1/24*(3*b^4*c^2*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*sq
rt(c)*x + a)^(5/2)*b^4*c^2 - 8*(b*sqrt(c)*x + a)^(3/2)*a*b^4*c^2 - 3*sqrt(b*sqrt
(c)*x + a)*a^2*b^4*c^2)/(a^2*b^3*c^(3/2)*x^3))/(b*sqrt(c))