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

Optimal. Leaf size=47 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{c x}{2 b \left (a+b x^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0452267, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{c x}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.95755, size = 39, normalized size = 0.83 \[ - \frac{c x}{2 b \left (a + b x^{2}\right )} + \frac{c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.03489, size = 47, normalized size = 1. \[ c \left (\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{x}{2 b \left (a+b x^2\right )}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[ -{\frac{cx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.236557, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{-a b} c x -{\left (b c x^{2} + a c\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{4 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{-a b}}, -\frac{\sqrt{a b} c x -{\left (b c x^{2} + a c\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.44071, size = 80, normalized size = 1.7 \[ c \left (- \frac{x}{2 a b + 2 b^{2} x^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(-x/(2*a*b + 2*b**2*x**2) - sqrt(-1/(a*b**3))*log(-a*b*sqrt(-1/(a*b**3)) + x)/
4 + sqrt(-1/(a*b**3))*log(a*b*sqrt(-1/(a*b**3)) + x)/4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227216, size = 50, normalized size = 1.06 \[ \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} - \frac{c x}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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