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

Optimal. Leaf size=36 \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0412215, antiderivative size = 36, 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 \[ -\frac{\sqrt{b} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{c}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.88082, size = 31, normalized size = 0.86 \[ - \frac{c}{a x} - \frac{\sqrt{b} c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0219419, size = 36, normalized size = 1. \[ c \left (-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))

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Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \[ -{\frac{bc}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{c}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248306, size = 1, normalized size = 0.03 \[ \left [\frac{c x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac{c x \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + c}{a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.32918, size = 66, normalized size = 1.83 \[ c \left (\frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x \right )}}{2} - \frac{1}{a x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221115, size = 42, normalized size = 1.17 \[ -\frac{b c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{c}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*c*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - c/(a*x)