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

Optimal. Leaf size=33 \[ \frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0426848, antiderivative size = 33, 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 \[ \frac{c x}{b}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.0641, size = 29, normalized size = 0.88 \[ - \frac{\sqrt{a} c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + \frac{c x}{b} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0153518, size = 33, normalized size = 1. \[ c \left (\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 0.9 \[{\frac{cx}{b}}-{\frac{ac}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238078, size = 1, normalized size = 0.03 \[ \left [\frac{c \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 2 \, c x}{2 \, b}, -\frac{c \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - c x}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.2243, size = 58, normalized size = 1.76 \[ c \left (\frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (- b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (b \sqrt{- \frac{a}{b^{3}}} + x \right )}}{2} + \frac{x}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22312, size = 38, normalized size = 1.15 \[ -\frac{a c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{c x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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