3.2934 \(\int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^6} \, dx\)

Optimal. Leaf size=219 \[ -\frac{7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{128 a^{9/2} x^5}+\frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5} \]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(5*x^5) + (7*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(240*a^2*x^
3) + (7*b^4*c^2*Sqrt[a + b*Sqrt[c*x^2]])/(128*a^4*x) - (b*(c*x^2)^(5/2)*Sqrt[a +
b*Sqrt[c*x^2]])/(40*a*c^2*x^9) - (7*b^3*(c*x^2)^(5/2)*Sqrt[a + b*Sqrt[c*x^2]])/
(192*a^3*c*x^7) - (7*b^5*(c*x^2)^(5/2)*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])
/(128*a^(9/2)*x^5)

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Rubi [A]  time = 0.266059, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{7 b^5 \left (c x^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{128 a^{9/2} x^5}+\frac{7 b^4 c^2 \sqrt{a+b \sqrt{c x^2}}}{128 a^4 x}-\frac{7 b^3 \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{192 a^3 c x^7}+\frac{7 b^2 c \sqrt{a+b \sqrt{c x^2}}}{240 a^2 x^3}-\frac{b \left (c x^2\right )^{5/2} \sqrt{a+b \sqrt{c x^2}}}{40 a c^2 x^9}-\frac{\sqrt{a+b \sqrt{c x^2}}}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^2]]/x^6,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^2]]/(5*x^5) + (7*b^2*c*Sqrt[a + b*Sqrt[c*x^2]])/(240*a^2*x^
3) + (7*b^4*c^2*Sqrt[a + b*Sqrt[c*x^2]])/(128*a^4*x) - (b*(c*x^2)^(5/2)*Sqrt[a +
b*Sqrt[c*x^2]])/(40*a*c^2*x^9) - (7*b^3*(c*x^2)^(5/2)*Sqrt[a + b*Sqrt[c*x^2]])/
(192*a^3*c*x^7) - (7*b^5*(c*x^2)^(5/2)*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/Sqrt[a]])
/(128*a^(9/2)*x^5)

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Rubi in Sympy [A]  time = 26.5256, size = 201, normalized size = 0.92 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{5 x^{5}} - \frac{b \left (c x^{2}\right )^{\frac{5}{2}} \sqrt{a + b \sqrt{c x^{2}}}}{40 a c^{2} x^{9}} + \frac{7 b^{2} c \sqrt{a + b \sqrt{c x^{2}}}}{240 a^{2} x^{3}} - \frac{7 b^{3} \left (c x^{2}\right )^{\frac{5}{2}} \sqrt{a + b \sqrt{c x^{2}}}}{192 a^{3} c x^{7}} + \frac{7 b^{4} c^{2} \sqrt{a + b \sqrt{c x^{2}}}}{128 a^{4} x} - \frac{7 b^{5} \left (c x^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{128 a^{\frac{9}{2}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**6,x)

[Out]

-sqrt(a + b*sqrt(c*x**2))/(5*x**5) - b*(c*x**2)**(5/2)*sqrt(a + b*sqrt(c*x**2))/
(40*a*c**2*x**9) + 7*b**2*c*sqrt(a + b*sqrt(c*x**2))/(240*a**2*x**3) - 7*b**3*(c
*x**2)**(5/2)*sqrt(a + b*sqrt(c*x**2))/(192*a**3*c*x**7) + 7*b**4*c**2*sqrt(a +
b*sqrt(c*x**2))/(128*a**4*x) - 7*b**5*(c*x**2)**(5/2)*atanh(sqrt(a + b*sqrt(c*x*
*2))/sqrt(a))/(128*a**(9/2)*x**5)

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Mathematica [A]  time = 0.0331099, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^6,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^6, x]

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Maple [A]  time = 0.021, size = 133, normalized size = 0.6 \[ -{\frac{1}{1920\,{x}^{5}} \left ( 105\,{a}^{17/2}\sqrt{a+b\sqrt{c{x}^{2}}}+790\,{a}^{15/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}-896\,{a}^{13/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}+490\,{a}^{11/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{7/2}-105\,{a}^{9/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{9/2}+105\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{4}{b}^{5} \left ( c{x}^{2} \right ) ^{5/2} \right ){a}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(1/2))^(1/2)/x^6,x)

[Out]

-1/1920*(105*a^(17/2)*(a+b*(c*x^2)^(1/2))^(1/2)+790*a^(15/2)*(a+b*(c*x^2)^(1/2))
^(3/2)-896*a^(13/2)*(a+b*(c*x^2)^(1/2))^(5/2)+490*a^(11/2)*(a+b*(c*x^2)^(1/2))^(
7/2)-105*a^(9/2)*(a+b*(c*x^2)^(1/2))^(9/2)+105*arctanh((a+b*(c*x^2)^(1/2))^(1/2)
/a^(1/2))*a^4*b^5*(c*x^2)^(5/2))/x^5/a^(17/2)

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227067, size = 1, normalized size = 0. \[ \left [\frac{105 \, b^{5} c^{2} x^{5} \sqrt{\frac{c}{a}} \log \left (\frac{\sqrt{c x^{2}} b c x + 2 \, a c x - 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} a \sqrt{\frac{c}{a}}}{x^{2}}\right ) + 2 \,{\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \,{\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3840 \, a^{4} x^{5}}, \frac{105 \, b^{5} c^{2} x^{5} \sqrt{-\frac{c}{a}} \arctan \left (\frac{a x \sqrt{-\frac{c}{a}}}{\sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a}}\right ) +{\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \,{\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{1920 \, a^{4} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^6,x, algorithm="fricas")

[Out]

[1/3840*(105*b^5*c^2*x^5*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*(105*b^4*c^2*x^4 + 56*a^2*b^2*
c*x^2 - 384*a^4 - 2*(35*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b
+ a))/(a^4*x^5), 1/1920*(105*b^5*c^2*x^5*sqrt(-c/a)*arctan(a*x*sqrt(-c/a)/(sqrt(
c*x^2)*sqrt(sqrt(c*x^2)*b + a))) + (105*b^4*c^2*x^4 + 56*a^2*b^2*c*x^2 - 384*a^4
- 2*(35*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^5)
]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**6,x)

[Out]

Integral(sqrt(a + b*sqrt(c*x**2))/x**6, x)

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GIAC/XCAS [A]  time = 0.22202, size = 211, normalized size = 0.96 \[ \frac{\frac{105 \, b^{6} c^{3} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} b^{6} c^{3} - 490 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a b^{6} c^{3} + 896 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{2} b^{6} c^{3} - 790 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{3} b^{6} c^{3} - 105 \, \sqrt{b \sqrt{c} x + a} a^{4} b^{6} c^{3}}{a^{4} b^{5} c^{\frac{5}{2}} x^{5}}}{1920 \, b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^2)*b + a)/x^6,x, algorithm="giac")

[Out]

1/1920*(105*b^6*c^3*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (105
*(b*sqrt(c)*x + a)^(9/2)*b^6*c^3 - 490*(b*sqrt(c)*x + a)^(7/2)*a*b^6*c^3 + 896*(
b*sqrt(c)*x + a)^(5/2)*a^2*b^6*c^3 - 790*(b*sqrt(c)*x + a)^(3/2)*a^3*b^6*c^3 - 1
05*sqrt(b*sqrt(c)*x + a)*a^4*b^6*c^3)/(a^4*b^5*c^(5/2)*x^5))/(b*sqrt(c))