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

Optimal. Leaf size=133 \[ -\frac{3 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{5/2}}+\frac{x \left (3 b^2-8 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a^2 \left (b^2-4 a c\right )}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.250395, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{3 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{5/2}}+\frac{x \left (3 b^2-8 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a^2 \left (b^2-4 a c\right )}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c/x^2 + b/x)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.2972, size = 117, normalized size = 0.88 \[ - \frac{2 x \left (- 2 a c + b^{2} + \frac{b c}{x}\right )}{a \left (- 4 a c + b^{2}\right ) \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} + \frac{x \left (- 16 a c + 6 b^{2}\right ) \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}}{2 a^{2} \left (- 4 a c + b^{2}\right )} - \frac{3 b \operatorname{atanh}{\left (\frac{2 a + \frac{b}{x}}{2 \sqrt{a} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x*(-2*a*c + b**2 + b*c/x)/(a*(-4*a*c + b**2)*sqrt(a + b/x + c/x**2)) + x*(-16
*a*c + 6*b**2)*sqrt(a + b/x + c/x**2)/(2*a**2*(-4*a*c + b**2)) - 3*b*atanh((2*a
+ b/x)/(2*sqrt(a)*sqrt(a + b/x + c/x**2)))/(2*a**(5/2))

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Mathematica [A]  time = 0.193161, size = 136, normalized size = 1.02 \[ -\frac{3 b \left (b^2-4 a c\right ) \sqrt{x (a x+b)+c} \log \left (2 \sqrt{a} \sqrt{x (a x+b)+c}+2 a x+b\right )+2 \sqrt{a} \left (-b^2 \left (a x^2+3 c\right )+10 a b c x+4 a c \left (a x^2+2 c\right )-3 b^3 x\right )}{2 a^{5/2} x \left (b^2-4 a c\right ) \sqrt{a+\frac{b x+c}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[a]*(-3*b^3*x + 10*a*b*c*x + 4*a*c*(2*c + a*x^2) - b^2*(3*c + a*x^2)) +
3*b*(b^2 - 4*a*c)*Sqrt[c + x*(b + a*x)]*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[c + x*(b
+ a*x)]])/(2*a^(5/2)*(b^2 - 4*a*c)*x*Sqrt[a + (c + b*x)/x^2])

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Maple [A]  time = 0.013, size = 197, normalized size = 1.5 \[ -{\frac{a{x}^{2}+bx+c}{2\,{x}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( -8\,{a}^{7/2}{x}^{2}c+2\,{a}^{5/2}{x}^{2}{b}^{2}-20\,{a}^{5/2}xbc+6\,{a}^{3/2}x{b}^{3}+12\,\sqrt{a{x}^{2}+bx+c}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}bc-3\,\sqrt{a{x}^{2}+bx+c}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3}-16\,{a}^{5/2}{c}^{2}+6\,{a}^{3/2}{b}^{2}c \right ){a}^{-{\frac{7}{2}}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)^(3/2),x)

[Out]

-1/2*(a*x^2+b*x+c)/a^(7/2)*(-8*a^(7/2)*x^2*c+2*a^(5/2)*x^2*b^2-20*a^(5/2)*x*b*c+
6*a^(3/2)*x*b^3+12*(a*x^2+b*x+c)^(1/2)*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*a^2*b*c-3*(a*x^2+b*x+c)^(1/2)*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/
2)+2*a*x+b)/a^(1/2))*a*b^3-16*a^(5/2)*c^2+6*a^(3/2)*b^2*c)/((a*x^2+b*x+c)/x^2)^(
3/2)/x^3/(4*a*c-b^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((a + b/x + c/x^2)^(-3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313032, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} c - 4 \, a b c^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt{a} \log \left (-{\left (8 \, a^{2} x^{2} + 8 \, a b x + b^{2} + 4 \, a c\right )} \sqrt{a} + 4 \,{\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}\right ) + 4 \,{\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}{4 \,{\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}, \frac{3 \,{\left (b^{3} c - 4 \, a b c^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt{-a} \arctan \left (\frac{{\left (2 \, a x + b\right )} \sqrt{-a}}{2 \, a x \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}\right ) + 2 \,{\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}{2 \,{\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(3*(b^3*c - 4*a*b*c^2 + (a*b^3 - 4*a^2*b*c)*x^2 + (b^4 - 4*a*b^2*c)*x)*sqrt
(a)*log(-(8*a^2*x^2 + 8*a*b*x + b^2 + 4*a*c)*sqrt(a) + 4*(2*a^2*x^2 + a*b*x)*sqr
t((a*x^2 + b*x + c)/x^2)) + 4*((a^2*b^2 - 4*a^3*c)*x^3 + (3*a*b^3 - 10*a^2*b*c)*
x^2 + (3*a*b^2*c - 8*a^2*c^2)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(a^3*b^2*c - 4*a^4
*c^2 + (a^4*b^2 - 4*a^5*c)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x), 1/2*(3*(b^3*c - 4*a*b
*c^2 + (a*b^3 - 4*a^2*b*c)*x^2 + (b^4 - 4*a*b^2*c)*x)*sqrt(-a)*arctan(1/2*(2*a*x
+ b)*sqrt(-a)/(a*x*sqrt((a*x^2 + b*x + c)/x^2))) + 2*((a^2*b^2 - 4*a^3*c)*x^3 +
(3*a*b^3 - 10*a^2*b*c)*x^2 + (3*a*b^2*c - 8*a^2*c^2)*x)*sqrt((a*x^2 + b*x + c)/
x^2))/(a^3*b^2*c - 4*a^4*c^2 + (a^4*b^2 - 4*a^5*c)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x
)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b/x + c/x**2)**(-3/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError