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

Optimal. Leaf size=25 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0226602, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b])

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Rubi in Sympy [A]  time = 5.43016, size = 24, normalized size = 0.96 \[ \frac{c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*atan(sqrt(b)*x/sqrt(a))/(sqrt(a)*sqrt(b))

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Mathematica [A]  time = 0.00701851, size = 25, normalized size = 1. \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.003, size = 17, normalized size = 0.7 \[{c\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229301, size = 1, normalized size = 0.04 \[ \left [\frac{c \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{2 \, \sqrt{-a b}}, \frac{c \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{\sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.327256, size = 54, normalized size = 2.16 \[ c \left (- \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x \right )}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22193, size = 22, normalized size = 0.88 \[ \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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