3.495 \(\int \frac{1}{x^5 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 b \sqrt{a+b x^2}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2}}{4 a x^4} \]

[Out]

-Sqrt[a + b*x^2]/(4*a*x^4) + (3*b*Sqrt[a + b*x^2])/(8*a^2*x^2) - (3*b^2*ArcTanh[
Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.113404, antiderivative size = 74, 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 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 b \sqrt{a+b x^2}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[a + b*x^2]),x]

[Out]

-Sqrt[a + b*x^2]/(4*a*x^4) + (3*b*Sqrt[a + b*x^2])/(8*a^2*x^2) - (3*b^2*ArcTanh[
Sqrt[a + b*x^2]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 10.7609, size = 66, normalized size = 0.89 \[ - \frac{\sqrt{a + b x^{2}}}{4 a x^{4}} + \frac{3 b \sqrt{a + b x^{2}}}{8 a^{2} x^{2}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**2)/(4*a*x**4) + 3*b*sqrt(a + b*x**2)/(8*a**2*x**2) - 3*b**2*atanh
(sqrt(a + b*x**2)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.0647646, size = 78, normalized size = 1.05 \[ \frac{-3 b^2 x^4 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} \sqrt{a+b x^2} \left (3 b x^2-2 a\right )+3 b^2 x^4 \log (x)}{8 a^{5/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[a + b*x^2]*(-2*a + 3*b*x^2) + 3*b^2*x^4*Log[x] - 3*b^2*x^4*Log[a +
Sqrt[a]*Sqrt[a + b*x^2]])/(8*a^(5/2)*x^4)

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Maple [A]  time = 0.01, size = 68, normalized size = 0.9 \[ -{\frac{1}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,b}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^2+a)^(1/2),x)

[Out]

-1/4*(b*x^2+a)^(1/2)/a/x^4+3/8*b*(b*x^2+a)^(1/2)/a^2/x^2-3/8*b^2/a^(5/2)*ln((2*a
+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241923, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \, b x^{2} - 2 \, a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, a^{\frac{5}{2}} x^{4}}, -\frac{3 \, b^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \, b x^{2} - 2 \, a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*b^2*x^4*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) + 2*(3*
b*x^2 - 2*a)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^4), -1/8*(3*b^2*x^4*arctan(sqrt
(-a)/sqrt(b*x^2 + a)) - (3*b*x^2 - 2*a)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*
x^4)]

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Sympy [A]  time = 13.6364, size = 97, normalized size = 1.31 \[ - \frac{1}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b}}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)/(8*a*x**3*sqrt(a/(b*x**2) + 1
)) + 3*b**(3/2)/(8*a**2*x*sqrt(a/(b*x**2) + 1)) - 3*b**2*asinh(sqrt(a)/(sqrt(b)*
x))/(8*a**(5/2))

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GIAC/XCAS [A]  time = 0.20579, size = 89, normalized size = 1.2 \[ \frac{1}{8} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 5 \, \sqrt{b x^{2} + a} a}{a^{2} b^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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