3.1454 \(\int \frac{1}{(2+3 x) \sqrt{1+5 x}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{5 x+1}\right )}{\sqrt{21}} \]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

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Rubi [A]  time = 0.0247372, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{5 x+1}\right )}{\sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + 3*x)*Sqrt[1 + 5*x]),x]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

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Rubi in Sympy [A]  time = 3.00296, size = 24, normalized size = 0.96 \[ \frac{2 \sqrt{21} \operatorname{atan}{\left (\frac{\sqrt{21} \sqrt{5 x + 1}}{7} \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*x)/(1+5*x)**(1/2),x)

[Out]

2*sqrt(21)*atan(sqrt(21)*sqrt(5*x + 1)/7)/21

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Mathematica [A]  time = 0.0201065, size = 25, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{5 x+1}\right )}{\sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 + 3*x)*Sqrt[1 + 5*x]),x]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

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Maple [A]  time = 0.009, size = 19, normalized size = 0.8 \[{\frac{2\,\sqrt{21}}{21}\arctan \left ({\frac{\sqrt{21}}{7}\sqrt{1+5\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+3*x)/(1+5*x)^(1/2),x)

[Out]

2/21*arctan(1/7*21^(1/2)*(1+5*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48735, size = 24, normalized size = 0.96 \[ \frac{2}{21} \, \sqrt{21} \arctan \left (\frac{1}{7} \, \sqrt{21} \sqrt{5 \, x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 1)*(3*x + 2)),x, algorithm="maxima")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))

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Fricas [A]  time = 0.221387, size = 24, normalized size = 0.96 \[ \frac{2}{21} \, \sqrt{21} \arctan \left (\frac{1}{7} \, \sqrt{21} \sqrt{5 \, x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 1)*(3*x + 2)),x, algorithm="fricas")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))

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Sympy [A]  time = 1.64547, size = 63, normalized size = 2.52 \[ \begin{cases} \frac{2 \sqrt{21} i \operatorname{acosh}{\left (\frac{\sqrt{105}}{15 \sqrt{x + \frac{2}{3}}} \right )}}{21} & \text{for}\: \frac{7 \left |{\frac{1}{x + \frac{2}{3}}}\right |}{15} > 1 \\- \frac{2 \sqrt{21} \operatorname{asin}{\left (\frac{\sqrt{105}}{15 \sqrt{x + \frac{2}{3}}} \right )}}{21} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*x)/(1+5*x)**(1/2),x)

[Out]

Piecewise((2*sqrt(21)*I*acosh(sqrt(105)/(15*sqrt(x + 2/3)))/21, 7*Abs(1/(x + 2/3
))/15 > 1), (-2*sqrt(21)*asin(sqrt(105)/(15*sqrt(x + 2/3)))/21, True))

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GIAC/XCAS [A]  time = 0.212859, size = 24, normalized size = 0.96 \[ \frac{2}{21} \, \sqrt{21} \arctan \left (\frac{1}{7} \, \sqrt{21} \sqrt{5 \, x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 1)*(3*x + 2)),x, algorithm="giac")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))