3.110 \(\int \frac{-x+4 x^3}{\left (5+x^2\right )^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]

[Out]

21/(2*(5 + x^2)) + 2*Log[5 + x^2]

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Rubi [A]  time = 0.049741, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-x + 4*x^3)/(5 + x^2)^2,x]

[Out]

21/(2*(5 + x^2)) + 2*Log[5 + x^2]

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Rubi in Sympy [A]  time = 9.17442, size = 15, normalized size = 0.75 \[ 2 \log{\left (x^{2} + 5 \right )} + \frac{21}{2 \left (x^{2} + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**3-x)/(x**2+5)**2,x)

[Out]

2*log(x**2 + 5) + 21/(2*(x**2 + 5))

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Mathematica [A]  time = 0.0126905, size = 20, normalized size = 1. \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-x + 4*x^3)/(5 + x^2)^2,x]

[Out]

21/(2*(5 + x^2)) + 2*Log[5 + x^2]

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Maple [A]  time = 0.014, size = 19, normalized size = 1. \[{\frac{21}{2\,{x}^{2}+10}}+2\,\ln \left ({x}^{2}+5 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^3-x)/(x^2+5)^2,x)

[Out]

21/2/(x^2+5)+2*ln(x^2+5)

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Maxima [A]  time = 1.34696, size = 24, normalized size = 1.2 \[ \frac{21}{2 \,{\left (x^{2} + 5\right )}} + 2 \, \log \left (x^{2} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x)/(x^2 + 5)^2,x, algorithm="maxima")

[Out]

21/2/(x^2 + 5) + 2*log(x^2 + 5)

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Fricas [A]  time = 0.223175, size = 32, normalized size = 1.6 \[ \frac{4 \,{\left (x^{2} + 5\right )} \log \left (x^{2} + 5\right ) + 21}{2 \,{\left (x^{2} + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x)/(x^2 + 5)^2,x, algorithm="fricas")

[Out]

1/2*(4*(x^2 + 5)*log(x^2 + 5) + 21)/(x^2 + 5)

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Sympy [A]  time = 0.097345, size = 15, normalized size = 0.75 \[ 2 \log{\left (x^{2} + 5 \right )} + \frac{21}{2 x^{2} + 10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**3-x)/(x**2+5)**2,x)

[Out]

2*log(x**2 + 5) + 21/(2*x**2 + 10)

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GIAC/XCAS [A]  time = 0.219991, size = 34, normalized size = 1.7 \[ -\frac{4 \, x^{2} - 1}{2 \,{\left (x^{2} + 5\right )}} + 2 \,{\rm ln}\left (x^{2} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x)/(x^2 + 5)^2,x, algorithm="giac")

[Out]

-1/2*(4*x^2 - 1)/(x^2 + 5) + 2*ln(x^2 + 5)