3.504 \(\int \frac{1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}+\frac{1}{a x^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.108735, antiderivative size = 68, 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 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 \sqrt{a+b x^2}}{2 a^2 x^2}+\frac{1}{a x^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.664, size = 63, normalized size = 0.93 \[ \frac{1}{a x^{2} \sqrt{a + b x^{2}}} - \frac{3 \sqrt{a + b x^{2}}}{2 a^{2} x^{2}} + \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.151314, size = 67, normalized size = 0.99 \[ \frac{-\frac{\sqrt{a} \left (a+3 b x^2\right )}{x^2 \sqrt{a+b x^2}}+3 b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-3 b \log (x)}{2 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 63, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,b}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)^(3/2),x)

[Out]

-1/2/a/x^2/(b*x^2+a)^(1/2)-3/2*b/a^2/(b*x^2+a)^(1/2)+3/2*b/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/((b*x^2 + a)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23984, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, b x^{2} + a\right )} \sqrt{b x^{2} + a} \sqrt{a} - 3 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{4 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{a}}, -\frac{{\left (3 \, b x^{2} + a\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.295, size = 73, normalized size = 1.07 \[ - \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**2+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215673, size = 89, normalized size = 1.31 \[ -\frac{1}{2} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \, b x^{2} + a}{{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{2} + a} a\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x^3),x, algorithm="giac")

[Out]

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