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

Optimal. Leaf size=73 \[ \frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.238015, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{c \sqrt{\frac{c}{a+b x^2}}}{a}-\frac{c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x^2))^(3/2)/x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.25011, size = 60, normalized size = 0.82 \[ \frac{c \sqrt{\frac{c}{a + b x^{2}}}}{a} - \frac{c \sqrt{\frac{c}{a + b x^{2}}} \sqrt{a + b x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0667933, size = 75, normalized size = 1.03 \[ \frac{c \sqrt{\frac{c}{a+b x^2}} \left (\log (x) \sqrt{a+b x^2}-\sqrt{a+b x^2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 64, normalized size = 0.9 \[ -{(b{x}^{2}+a) \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}} \left ( \ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ) a\sqrt{b{x}^{2}+a}-{a}^{{\frac{3}{2}}} \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x^2+a))^(3/2)/x,x)

[Out]

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

_______________________________________________________________________________________

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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.287349, size = 1, normalized size = 0.01 \[ \left [\frac{c \sqrt{\frac{c}{a}} \log \left (-\frac{b c x^{2} + 2 \, a c - 2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{\frac{c}{a}}}{x^{2}}\right ) + 2 \, c \sqrt{\frac{c}{b x^{2} + a}}}{2 \, a}, -\frac{c \sqrt{-\frac{c}{a}} \arctan \left (\frac{c}{{\left (b x^{2} + a\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{-\frac{c}{a}}}\right ) - c \sqrt{\frac{c}{b x^{2} + a}}}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272139, size = 88, normalized size = 1.21 \[ c^{3}{\left (\frac{\arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a c} + \frac{1}{\sqrt{b c x^{2} + a c} a c}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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