3.2928 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^5} \, dx\)

Optimal. Leaf size=171 \[ \frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{64 a^{7/2}}-\frac{5 b^3 c^2 \sqrt{a+b \sqrt{c x^2}}}{64 a^3 \sqrt{c x^2}}+\frac{5 b^2 c \sqrt{a+b \sqrt{c x^2}}}{96 a^2 x^2}-\frac{b c^2 \sqrt{a+b \sqrt{c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{4 x^4} \]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(4*x^4) + (5*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(96*a^2*x^2
) - (b*c^2*Sqrt[a + b*Sqrt[c*x^2]])/(24*a*(c*x^2)^(3/2)) - (5*b^3*c^2*Sqrt[a + b
*Sqrt[c*x^2]])/(64*a^3*Sqrt[c*x^2]) + (5*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]
/Sqrt[a]])/(64*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.218376, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{64 a^{7/2}}-\frac{5 b^3 c^2 \sqrt{a+b \sqrt{c x^2}}}{64 a^3 \sqrt{c x^2}}+\frac{5 b^2 c \sqrt{a+b \sqrt{c x^2}}}{96 a^2 x^2}-\frac{b c^2 \sqrt{a+b \sqrt{c x^2}}}{24 a \left (c x^2\right )^{3/2}}-\frac{\sqrt{a+b \sqrt{c x^2}}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^2]]/x^5,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(4*x^4) + (5*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(96*a^2*x^2
) - (b*c^2*Sqrt[a + b*Sqrt[c*x^2]])/(24*a*(c*x^2)^(3/2)) - (5*b^3*c^2*Sqrt[a + b
*Sqrt[c*x^2]])/(64*a^3*Sqrt[c*x^2]) + (5*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]
/Sqrt[a]])/(64*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.4676, size = 156, normalized size = 0.91 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{4 x^{4}} - \frac{b c^{2} \sqrt{a + b \sqrt{c x^{2}}}}{24 a \left (c x^{2}\right )^{\frac{3}{2}}} + \frac{5 b^{2} c \sqrt{a + b \sqrt{c x^{2}}}}{96 a^{2} x^{2}} - \frac{5 b^{3} c^{2} \sqrt{a + b \sqrt{c x^{2}}}}{64 a^{3} \sqrt{c x^{2}}} + \frac{5 b^{4} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{64 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**5,x)

[Out]

-sqrt(a + b*sqrt(c*x**2))/(4*x**4) - b*c**2*sqrt(a + b*sqrt(c*x**2))/(24*a*(c*x*
*2)**(3/2)) + 5*b**2*c*sqrt(a + b*sqrt(c*x**2))/(96*a**2*x**2) - 5*b**3*c**2*sqr
t(a + b*sqrt(c*x**2))/(64*a**3*sqrt(c*x**2)) + 5*b**4*c**2*atanh(sqrt(a + b*sqrt
(c*x**2))/sqrt(a))/(64*a**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0325477, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^5,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^5, x]

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 114, normalized size = 0.7 \[ -{\frac{1}{192\,{x}^{4}} \left ( 15\,{a}^{7/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{7/2}-15\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{3}{b}^{4}{c}^{2}{x}^{4}-55\,{a}^{9/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}+73\,{a}^{11/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}+15\,{a}^{13/2}\sqrt{a+b\sqrt{c{x}^{2}}} \right ){a}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(1/2))^(1/2)/x^5,x)

[Out]

-1/192*(15*a^(7/2)*(a+b*(c*x^2)^(1/2))^(7/2)-15*arctanh((a+b*(c*x^2)^(1/2))^(1/2
)/a^(1/2))*a^3*b^4*c^2*x^4-55*a^(9/2)*(a+b*(c*x^2)^(1/2))^(5/2)+73*a^(11/2)*(a+b
*(c*x^2)^(1/2))^(3/2)+15*a^(13/2)*(a+b*(c*x^2)^(1/2))^(1/2))/a^(13/2)/x^4

_______________________________________________________________________________________

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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225364, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{a} b^{4} c^{2} x^{4} \log \left (\frac{\sqrt{c x^{2}} \sqrt{a} b + 2 \, \sqrt{\sqrt{c x^{2}} b + a} a + 2 \, a^{\frac{3}{2}}}{x}\right ) + 2 \,{\left (10 \, a^{2} b^{2} c x^{2} - 48 \, a^{4} -{\left (15 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{384 \, a^{4} x^{4}}, \frac{15 \, \sqrt{-a} b^{4} c^{2} x^{4} \arctan \left (\frac{a}{\sqrt{\sqrt{c x^{2}} b + a} \sqrt{-a}}\right ) +{\left (10 \, a^{2} b^{2} c x^{2} - 48 \, a^{4} -{\left (15 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{192 \, a^{4} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^5,x, algorithm="fricas")

[Out]

[1/384*(15*sqrt(a)*b^4*c^2*x^4*log((sqrt(c*x^2)*sqrt(a)*b + 2*sqrt(sqrt(c*x^2)*b
+ a)*a + 2*a^(3/2))/x) + 2*(10*a^2*b^2*c*x^2 - 48*a^4 - (15*a*b^3*c*x^2 + 8*a^3
*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^4), 1/192*(15*sqrt(-a)*b^4*c^2*
x^4*arctan(a/(sqrt(sqrt(c*x^2)*b + a)*sqrt(-a))) + (10*a^2*b^2*c*x^2 - 48*a^4 -
(15*a*b^3*c*x^2 + 8*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^4)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**5, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221841, size = 182, normalized size = 1.06 \[ -\frac{\frac{15 \, b^{5} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} b^{5} c^{\frac{5}{2}} - 55 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a b^{5} c^{\frac{5}{2}} + 73 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{2} b^{5} c^{\frac{5}{2}} + 15 \, \sqrt{b \sqrt{c} x + a} a^{3} b^{5} c^{\frac{5}{2}}}{a^{3} b^{4} c^{2} x^{4}}}{192 \, b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^5,x, algorithm="giac")

[Out]

-1/192*(15*b^5*c^(5/2)*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (
15*(b*sqrt(c)*x + a)^(7/2)*b^5*c^(5/2) - 55*(b*sqrt(c)*x + a)^(5/2)*a*b^5*c^(5/2
) + 73*(b*sqrt(c)*x + a)^(3/2)*a^2*b^5*c^(5/2) + 15*sqrt(b*sqrt(c)*x + a)*a^3*b^
5*c^(5/2))/(a^3*b^4*c^2*x^4))/(b*sqrt(c))