3.415 \(\int \frac{x}{4 \sqrt{x}+x} \, dx\)

Optimal. Leaf size=19 \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

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Rubi [A]  time = 0.0287146, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(4*Sqrt[x] + x),x]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 8 \sqrt{x} + 32 \log{\left (\sqrt{x} + 4 \right )} + 2 \int ^{\sqrt{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x+4*x**(1/2)),x)

[Out]

-8*sqrt(x) + 32*log(sqrt(x) + 4) + 2*Integral(x, (x, sqrt(x)))

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Mathematica [A]  time = 0.0089928, size = 19, normalized size = 1. \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(4*Sqrt[x] + x),x]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[ x+32\,\ln \left ( 4+\sqrt{x} \right ) -8\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x+4*x^(1/2)),x)

[Out]

x+32*ln(4+x^(1/2))-8*x^(1/2)

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Maxima [A]  time = 0.690106, size = 20, normalized size = 1.05 \[ x - 8 \, \sqrt{x} + 32 \, \log \left (\sqrt{x} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + 4*sqrt(x)),x, algorithm="maxima")

[Out]

x - 8*sqrt(x) + 32*log(sqrt(x) + 4)

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Fricas [A]  time = 0.26943, size = 20, normalized size = 1.05 \[ x - 8 \, \sqrt{x} + 32 \, \log \left (\sqrt{x} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + 4*sqrt(x)),x, algorithm="fricas")

[Out]

x - 8*sqrt(x) + 32*log(sqrt(x) + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{4 \sqrt{x} + x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x+4*x**(1/2)),x)

[Out]

Integral(x/(4*sqrt(x) + x), x)

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GIAC/XCAS [A]  time = 0.278019, size = 20, normalized size = 1.05 \[ x - 8 \, \sqrt{x} + 32 \,{\rm ln}\left (\sqrt{x} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + 4*sqrt(x)),x, algorithm="giac")

[Out]

x - 8*sqrt(x) + 32*ln(sqrt(x) + 4)