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

Optimal. Leaf size=17 \[ -\frac{c}{2 b \left (a+b x^2\right )} \]

[Out]

-c/(2*b*(a + b*x^2))

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Rubi [A]  time = 0.0131958, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{c}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*b*(a + b*x^2))

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Rubi in Sympy [A]  time = 4.76808, size = 12, normalized size = 0.71 \[ - \frac{c}{2 b \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(2*b*(a + b*x**2))

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Mathematica [A]  time = 0.00378476, size = 17, normalized size = 1. \[ -\frac{c}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*b*(a + b*x^2))

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[ -{\frac{c}{2\,b \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34255, size = 22, normalized size = 1.29 \[ -\frac{c}{2 \,{\left (b^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/(b^2*x^2 + a*b)

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Fricas [A]  time = 0.211298, size = 22, normalized size = 1.29 \[ -\frac{c}{2 \,{\left (b^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/(b^2*x^2 + a*b)

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Sympy [A]  time = 1.29719, size = 15, normalized size = 0.88 \[ - \frac{c}{2 a b + 2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(2*a*b + 2*b**2*x**2)

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GIAC/XCAS [A]  time = 0.230283, size = 20, normalized size = 1.18 \[ -\frac{c}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/((b*x^2 + a)*b)