3.2098 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2} x} \, dx\)

Optimal. Leaf size=64 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-1/(6*a*(a + b/x^4)^(3/2)) - 1/(2*a^2*Sqrt[a + b/x^4]) + ArcTanh[Sqrt[a + b/x^4]
/Sqrt[a]]/(2*a^(5/2))

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Rubi [A]  time = 0.115789, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(5/2)*x),x]

[Out]

-1/(6*a*(a + b/x^4)^(3/2)) - 1/(2*a^2*Sqrt[a + b/x^4]) + ArcTanh[Sqrt[a + b/x^4]
/Sqrt[a]]/(2*a^(5/2))

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Rubi in Sympy [A]  time = 9.73753, size = 54, normalized size = 0.84 \[ - \frac{1}{6 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{1}{2 a^{2} \sqrt{a + \frac{b}{x^{4}}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(5/2)/x,x)

[Out]

-1/(6*a*(a + b/x**4)**(3/2)) - 1/(2*a**2*sqrt(a + b/x**4)) + atanh(sqrt(a + b/x*
*4)/sqrt(a))/(2*a**(5/2))

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Mathematica [A]  time = 0.0675436, size = 90, normalized size = 1.41 \[ \frac{3 \left (a x^4+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )-\sqrt{a} x^2 \left (4 a x^4+3 b\right )}{6 a^{5/2} x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(5/2)*x),x]

[Out]

(-(Sqrt[a]*x^2*(3*b + 4*a*x^4)) + 3*(b + a*x^4)^(3/2)*Log[a*x^2 + Sqrt[a]*Sqrt[b
+ a*x^4]])/(6*a^(5/2)*Sqrt[a + b/x^4]*x^2*(b + a*x^4))

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Maple [B]  time = 0.036, size = 221, normalized size = 3.5 \[{\frac{1}{6\,{x}^{10}} \left ( a{x}^{4}+b \right ) ^{{\frac{5}{2}}} \left ( 3\,\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{8}{a}^{5}-4\,\sqrt{-{\frac{ \left ( a{x}^{2}+\sqrt{-ab} \right ) \left ( -a{x}^{2}+\sqrt{-ab} \right ) }{a}}}{a}^{9/2}{x}^{6}+6\,\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){a}^{4}b{x}^{4}-3\,\sqrt{-{\frac{ \left ( a{x}^{2}+\sqrt{-ab} \right ) \left ( -a{x}^{2}+\sqrt{-ab} \right ) }{a}}}{a}^{7/2}b{x}^{2}+3\,\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){a}^{3}{b}^{2} \right ){a}^{-{\frac{7}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}} \left ( -a{x}^{2}+\sqrt{-ab} \right ) ^{-2} \left ( a{x}^{2}+\sqrt{-ab} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(5/2)/x,x)

[Out]

1/6*(a*x^4+b)^(5/2)/a^(7/2)*(3*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^8*a^5-4*(-1/a*(
a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^(1/2)))^(1/2)*a^(9/2)*x^6+6*ln(x^2*a^(1/2)+(a
*x^4+b)^(1/2))*a^4*b*x^4-3*(-1/a*(a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^(1/2)))^(1/
2)*a^(7/2)*b*x^2+3*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*a^3*b^2)/((a*x^4+b)/x^4)^(5/2
)/x^10/(-a*x^2+(-a*b)^(1/2))^2/(a*x^2+(-a*b)^(1/2))^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(1/((a + b/x^4)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261122, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{a} \log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right ) - 2 \,{\left (4 \, a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \,{\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}}, -\frac{3 \,{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (4 \, a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \,{\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/12*(3*(a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt(a)*log(-2*a*x^4*sqrt((a*x^4 + b)/x^4)
- (2*a*x^4 + b)*sqrt(a)) - 2*(4*a^2*x^8 + 3*a*b*x^4)*sqrt((a*x^4 + b)/x^4))/(a^5
*x^8 + 2*a^4*b*x^4 + a^3*b^2), -1/6*(3*(a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt(-a)*arct
an(sqrt(-a)/sqrt((a*x^4 + b)/x^4)) + (4*a^2*x^8 + 3*a*b*x^4)*sqrt((a*x^4 + b)/x^
4))/(a^5*x^8 + 2*a^4*b*x^4 + a^3*b^2)]

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Sympy [A]  time = 15.4389, size = 743, normalized size = 11.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(5/2)/x,x)

[Out]

-8*a**7*x**12*sqrt(1 + b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 3
6*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 3*a**7*x**12*log(b/(a*x**4))/(12*a*
*(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3
) + 6*a**7*x**12*log(sqrt(1 + b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36*a**(17/2
)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 14*a**6*b*x**8*sqrt(1 +
b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4
+ 12*a**(13/2)*b**3) - 9*a**6*b*x**8*log(b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a*
*(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) + 18*a**6*b*x**8*lo
g(sqrt(1 + b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(1
5/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 6*a**5*b**2*x**4*sqrt(1 + b/(a*x**4))/(12*
a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b*
*3) - 9*a**5*b**2*x**4*log(b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8
+ 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) + 18*a**5*b**2*x**4*log(sqrt(1 +
b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x
**4 + 12*a**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x**4))/(12*a**(19/2)*x**12 + 36*
a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) + 6*a**4*b**3*log
(sqrt(1 + b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15
/2)*b**2*x**4 + 12*a**(13/2)*b**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(5/2)*x), x)