3.1954 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^8} \, dx\)

Optimal. Leaf size=95 \[ \frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x^2}}}{2 b^3 x}+\frac{5}{3 b^2 x^3 \sqrt{a+\frac{b}{x^2}}}+\frac{1}{3 b x^5 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

1/(3*b*(a + b/x^2)^(3/2)*x^5) + 5/(3*b^2*Sqrt[a + b/x^2]*x^3) - (5*Sqrt[a + b/x^
2])/(2*b^3*x) + (5*a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.148155, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x^2}}}{2 b^3 x}+\frac{5}{3 b^2 x^3 \sqrt{a+\frac{b}{x^2}}}+\frac{1}{3 b x^5 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(5/2)*x^8),x]

[Out]

1/(3*b*(a + b/x^2)^(3/2)*x^5) + 5/(3*b^2*Sqrt[a + b/x^2]*x^3) - (5*Sqrt[a + b/x^
2])/(2*b^3*x) + (5*a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(2*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7528, size = 85, normalized size = 0.89 \[ \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 b^{\frac{7}{2}}} + \frac{1}{3 b x^{5} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} + \frac{5}{3 b^{2} x^{3} \sqrt{a + \frac{b}{x^{2}}}} - \frac{5 \sqrt{a + \frac{b}{x^{2}}}}{2 b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x**8,x)

[Out]

5*a*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(2*b**(7/2)) + 1/(3*b*x**5*(a + b/x**2)*
*(3/2)) + 5/(3*b**2*x**3*sqrt(a + b/x**2)) - 5*sqrt(a + b/x**2)/(2*b**3*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.130607, size = 117, normalized size = 1.23 \[ \frac{-\sqrt{b} \left (15 a^2 x^4+20 a b x^2+3 b^2\right )-15 a x^2 \log (x) \left (a x^2+b\right )^{3/2}+15 a x^2 \left (a x^2+b\right )^{3/2} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{6 b^{7/2} x^3 \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(5/2)*x^8),x]

[Out]

(-(Sqrt[b]*(3*b^2 + 20*a*b*x^2 + 15*a^2*x^4)) - 15*a*x^2*(b + a*x^2)^(3/2)*Log[x
] + 15*a*x^2*(b + a*x^2)^(3/2)*Log[b + Sqrt[b]*Sqrt[b + a*x^2]])/(6*b^(7/2)*Sqrt
[a + b/x^2]*x^3*(b + a*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 92, normalized size = 1. \[ -{\frac{a{x}^{2}+b}{6\,{x}^{7}} \left ( 15\,{b}^{3/2}{x}^{4}{a}^{2}+20\,{b}^{5/2}{x}^{2}a-15\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{2}ab+3\,{b}^{7/2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^8,x)

[Out]

-1/6*(a*x^2+b)*(15*b^(3/2)*x^4*a^2+20*b^(5/2)*x^2*a-15*ln(2*(b^(1/2)*(a*x^2+b)^(
1/2)+b)/x)*(a*x^2+b)^(3/2)*x^2*a*b+3*b^(7/2))/((a*x^2+b)/x^2)^(5/2)/x^7/b^(9/2)

_______________________________________________________________________________________

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(1/((a + b/x^2)^(5/2)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.261959, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{3} x^{5} + 2 \, a^{2} b x^{3} + a b^{2} x\right )} \sqrt{b} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (15 \, a^{2} b x^{4} + 20 \, a b^{2} x^{2} + 3 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \,{\left (a^{2} b^{4} x^{5} + 2 \, a b^{5} x^{3} + b^{6} x\right )}}, -\frac{15 \,{\left (a^{3} x^{5} + 2 \, a^{2} b x^{3} + a b^{2} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (15 \, a^{2} b x^{4} + 20 \, a b^{2} x^{2} + 3 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \,{\left (a^{2} b^{4} x^{5} + 2 \, a b^{5} x^{3} + b^{6} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^8),x, algorithm="fricas")

[Out]

[1/12*(15*(a^3*x^5 + 2*a^2*b*x^3 + a*b^2*x)*sqrt(b)*log(-(2*b*x*sqrt((a*x^2 + b)
/x^2) + (a*x^2 + 2*b)*sqrt(b))/x^2) - 2*(15*a^2*b*x^4 + 20*a*b^2*x^2 + 3*b^3)*sq
rt((a*x^2 + b)/x^2))/(a^2*b^4*x^5 + 2*a*b^5*x^3 + b^6*x), -1/6*(15*(a^3*x^5 + 2*
a^2*b*x^3 + a*b^2*x)*sqrt(-b)*arctan(sqrt(-b)/(x*sqrt((a*x^2 + b)/x^2))) + (15*a
^2*b*x^4 + 20*a*b^2*x^2 + 3*b^3)*sqrt((a*x^2 + b)/x^2))/(a^2*b^4*x^5 + 2*a*b^5*x
^3 + b^6*x)]

_______________________________________________________________________________________

Sympy [A]  time = 35.2349, size = 864, normalized size = 9.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(5/2)/x**8,x)

[Out]

-15*a**4*b**13*x**8*log(a*x**2/b)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x*
*6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) + 30*a**4*b**13*x**8*log(sqrt(a*x*
*2/b + 1) + 1)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)
*x**4 + 12*b**(39/2)*x**2) - 30*a**3*b**14*x**6*sqrt(a*x**2/b + 1)/(12*a**3*b**(
33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) -
45*a**3*b**14*x**6*log(a*x**2/b)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x*
*6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) + 90*a**3*b**14*x**6*log(sqrt(a*x*
*2/b + 1) + 1)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)
*x**4 + 12*b**(39/2)*x**2) - 70*a**2*b**15*x**4*sqrt(a*x**2/b + 1)/(12*a**3*b**(
33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) -
45*a**2*b**15*x**4*log(a*x**2/b)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x*
*6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) + 90*a**2*b**15*x**4*log(sqrt(a*x*
*2/b + 1) + 1)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)
*x**4 + 12*b**(39/2)*x**2) - 46*a*b**16*x**2*sqrt(a*x**2/b + 1)/(12*a**3*b**(33/
2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) - 15
*a*b**16*x**2*log(a*x**2/b)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 3
6*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2) + 30*a*b**16*x**2*log(sqrt(a*x**2/b + 1)
+ 1)/(12*a**3*b**(33/2)*x**8 + 36*a**2*b**(35/2)*x**6 + 36*a*b**(37/2)*x**4 + 1
2*b**(39/2)*x**2) - 6*b**17*sqrt(a*x**2/b + 1)/(12*a**3*b**(33/2)*x**8 + 36*a**2
*b**(35/2)*x**6 + 36*a*b**(37/2)*x**4 + 12*b**(39/2)*x**2)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^8),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x^8), x)