3.1904 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=95 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{16 b x}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{6 x^3} \]

[Out]

-(a*Sqrt[a + b/x^2])/(8*x^3) - (a + b/x^2)^(3/2)/(6*x^3) - (a^2*Sqrt[a + b/x^2])
/(16*b*x) + (a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(16*b^(3/2))

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Rubi [A]  time = 0.143646, 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 \[ \frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 b^{3/2}}-\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{16 b x}-\frac{a \sqrt{a+\frac{b}{x^2}}}{8 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)/x^4,x]

[Out]

-(a*Sqrt[a + b/x^2])/(8*x^3) - (a + b/x^2)^(3/2)/(6*x^3) - (a^2*Sqrt[a + b/x^2])
/(16*b*x) + (a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(16*b^(3/2))

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Rubi in Sympy [A]  time = 14.1463, size = 78, normalized size = 0.82 \[ \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{16 b^{\frac{3}{2}}} - \frac{a^{2} \sqrt{a + \frac{b}{x^{2}}}}{16 b x} - \frac{a \sqrt{a + \frac{b}{x^{2}}}}{8 x^{3}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(3/2)/x**4,x)

[Out]

a**3*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(16*b**(3/2)) - a**2*sqrt(a + b/x**2)/(
16*b*x) - a*sqrt(a + b/x**2)/(8*x**3) - (a + b/x**2)**(3/2)/(6*x**3)

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Mathematica [A]  time = 0.1085, size = 111, normalized size = 1.17 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (-3 a^3 x^6 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+3 a^3 x^6 \log (x)+\sqrt{b} \sqrt{a x^2+b} \left (3 a^2 x^4+14 a b x^2+8 b^2\right )\right )}{48 b^{3/2} x^5 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(3/2)/x^4,x]

[Out]

-(Sqrt[a + b/x^2]*(Sqrt[b]*Sqrt[b + a*x^2]*(8*b^2 + 14*a*b*x^2 + 3*a^2*x^4) + 3*
a^3*x^6*Log[x] - 3*a^3*x^6*Log[b + Sqrt[b]*Sqrt[b + a*x^2]]))/(48*b^(3/2)*x^5*Sq
rt[b + a*x^2])

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Maple [A]  time = 0.016, size = 145, normalized size = 1.5 \[{\frac{1}{48\,{b}^{3}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{6}{a}^{3}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{6}{a}^{3}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{4}{a}^{2}-3\,\sqrt{a{x}^{2}+b}{x}^{6}{a}^{3}b+2\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}ab-8\, \left ( a{x}^{2}+b \right ) ^{5/2}{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)/x^4,x)

[Out]

1/48*((a*x^2+b)/x^2)^(3/2)/x^3*(3*b^(3/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^
6*a^3-(a*x^2+b)^(3/2)*x^6*a^3+(a*x^2+b)^(5/2)*x^4*a^2-3*(a*x^2+b)^(1/2)*x^6*a^3*
b+2*(a*x^2+b)^(5/2)*x^2*a*b-8*(a*x^2+b)^(5/2)*b^2)/(a*x^2+b)^(3/2)/b^3

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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((a + b/x^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258503, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \sqrt{b} x^{5} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{96 \, b^{2} x^{5}}, -\frac{3 \, a^{3} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a^{2} b x^{4} + 14 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{48 \, b^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[1/96*(3*a^3*sqrt(b)*x^5*log(-(2*b*x*sqrt((a*x^2 + b)/x^2) + (a*x^2 + 2*b)*sqrt(
b))/x^2) - 2*(3*a^2*b*x^4 + 14*a*b^2*x^2 + 8*b^3)*sqrt((a*x^2 + b)/x^2))/(b^2*x^
5), -1/48*(3*a^3*sqrt(-b)*x^5*arctan(sqrt(-b)/(x*sqrt((a*x^2 + b)/x^2))) + (3*a^
2*b*x^4 + 14*a*b^2*x^2 + 8*b^3)*sqrt((a*x^2 + b)/x^2))/(b^2*x^5)]

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Sympy [A]  time = 18.8757, size = 119, normalized size = 1.25 \[ - \frac{a^{\frac{5}{2}}}{16 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{17 a^{\frac{3}{2}}}{48 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{11 \sqrt{a} b}{24 x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{16 b^{\frac{3}{2}}} - \frac{b^{2}}{6 \sqrt{a} x^{7} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(3/2)/x**4,x)

[Out]

-a**(5/2)/(16*b*x*sqrt(1 + b/(a*x**2))) - 17*a**(3/2)/(48*x**3*sqrt(1 + b/(a*x**
2))) - 11*sqrt(a)*b/(24*x**5*sqrt(1 + b/(a*x**2))) + a**3*asinh(sqrt(b)/(sqrt(a)
*x))/(16*b**(3/2)) - b**2/(6*sqrt(a)*x**7*sqrt(1 + b/(a*x**2)))

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GIAC/XCAS [A]  time = 0.264781, size = 111, normalized size = 1.17 \[ -\frac{1}{48} \, a^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} + 8 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{a x^{2} + b} b^{2}}{a^{3} b x^{6}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

-1/48*a^3*(3*arctan(sqrt(a*x^2 + b)/sqrt(-b))/(sqrt(-b)*b) + (3*(a*x^2 + b)^(5/2
) + 8*(a*x^2 + b)^(3/2)*b - 3*sqrt(a*x^2 + b)*b^2)/(a^3*b*x^6))*sign(x)