3.404 \(\int \frac{1}{\sqrt{x}+4 x^{3/2}} \, dx\)

Optimal. Leaf size=8 \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

[Out]

ArcTan[2*Sqrt[x]]

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Rubi [A]  time = 0.0102897, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[x] + 4*x^(3/2))^(-1),x]

[Out]

ArcTan[2*Sqrt[x]]

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Rubi in Sympy [A]  time = 1.32601, size = 7, normalized size = 0.88 \[ \operatorname{atan}{\left (2 \sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**(3/2)+x**(1/2)),x)

[Out]

atan(2*sqrt(x))

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Mathematica [A]  time = 0.00550019, size = 8, normalized size = 1. \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[x] + 4*x^(3/2))^(-1),x]

[Out]

ArcTan[2*Sqrt[x]]

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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \[ \arctan \left ( 2\,\sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^(3/2)+x^(1/2)),x)

[Out]

arctan(2*x^(1/2))

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Maxima [A]  time = 0.796948, size = 8, normalized size = 1. \[ \arctan \left (2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x^(3/2) + sqrt(x)),x, algorithm="maxima")

[Out]

arctan(2*sqrt(x))

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Fricas [A]  time = 0.271161, size = 8, normalized size = 1. \[ \arctan \left (2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x^(3/2) + sqrt(x)),x, algorithm="fricas")

[Out]

arctan(2*sqrt(x))

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Sympy [A]  time = 0.555798, size = 7, normalized size = 0.88 \[ \operatorname{atan}{\left (2 \sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**(3/2)+x**(1/2)),x)

[Out]

atan(2*sqrt(x))

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GIAC/XCAS [A]  time = 0.277453, size = 8, normalized size = 1. \[ \arctan \left (2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x^(3/2) + sqrt(x)),x, algorithm="giac")

[Out]

arctan(2*sqrt(x))