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

Optimal. Leaf size=112 \[ \frac{3 b c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 c \left (a+b x^2\right ) \sqrt{\frac{c}{a+b x^2}}}{2 a^2 x^2}+\frac{c \sqrt{\frac{c}{a+b x^2}}}{a x^2} \]

[Out]

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

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Rubi [A]  time = 0.275438, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{3 b c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 c \left (a+b x^2\right ) \sqrt{\frac{c}{a+b x^2}}}{2 a^2 x^2}+\frac{c \sqrt{\frac{c}{a+b x^2}}}{a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4834, size = 99, normalized size = 0.88 \[ \frac{c \sqrt{\frac{c}{a + b x^{2}}}}{a x^{2}} - \frac{3 c \sqrt{\frac{c}{a + b x^{2}}} \left (a + b x^{2}\right )}{2 a^{2} x^{2}} + \frac{3 b c \sqrt{\frac{c}{a + b x^{2}}} \sqrt{a + b x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0890055, size = 99, normalized size = 0.88 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}} \left (\sqrt{a} \left (a+3 b x^2\right )+3 b x^2 \log (x) \sqrt{a+b x^2}-3 b x^2 \sqrt{a+b x^2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )\right )}{2 a^{5/2} x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 79, normalized size = 0.7 \[ -{\frac{b{x}^{2}+a}{2\,{x}^{2}} \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{3/2}{x}^{2}b-3\,\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ) \sqrt{b{x}^{2}+a}{x}^{2}ab+{a}^{{\frac{5}{2}}} \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x^2+a))^(3/2)/x^3,x)

[Out]

-1/2*(c/(b*x^2+a))^(3/2)*(b*x^2+a)*(3*a^(3/2)*x^2*b-3*ln(2*(a^(1/2)*(b*x^2+a)^(1
/2)+a)/x)*(b*x^2+a)^(1/2)*x^2*a*b+a^(5/2))/a^(7/2)/x^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((c/(b*x^2 + a))^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296741, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b c x^{2} \sqrt{\frac{c}{a}} \log \left (-\frac{b c x^{2} + 2 \, a c + 2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{\frac{c}{a}}}{x^{2}}\right ) - 2 \,{\left (3 \, b c x^{2} + a c\right )} \sqrt{\frac{c}{b x^{2} + a}}}{4 \, a^{2} x^{2}}, \frac{3 \, b c x^{2} \sqrt{-\frac{c}{a}} \arctan \left (\frac{c}{{\left (b x^{2} + a\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{-\frac{c}{a}}}\right ) -{\left (3 \, b c x^{2} + a c\right )} \sqrt{\frac{c}{b x^{2} + a}}}{2 \, a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272278, size = 135, normalized size = 1.21 \[ -\frac{1}{2} \, b c^{4}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a^{2} c^{2}} - \frac{3 \, b c x^{2} + a c}{{\left (\sqrt{b c x^{2} + a c} a c -{\left (b c x^{2} + a c\right )}^{\frac{3}{2}}\right )} a^{2} c^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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