3.1311 \(\int \frac{(1+2 x)^{3/2}}{1+x+x^2} \, dx\)

Optimal. Leaf size=168 \[ 4 \sqrt{2 x+1}+\frac{\sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-\frac{\sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

[Out]

4*Sqrt[1 + 2*x] + Sqrt[2]*3^(1/4)*ArcTan[1 - (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)] -
Sqrt[2]*3^(1/4)*ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)] + (3^(1/4)*Log[1 + S
qrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/Sqrt[2] - (3^(1/4)*Log[1 + Sqrt[3
] + 2*x + Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/Sqrt[2]

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Rubi [A]  time = 0.303962, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ 4 \sqrt{2 x+1}+\frac{\sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-\frac{\sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*Sqrt[1 + 2*x] + Sqrt[2]*3^(1/4)*ArcTan[1 - (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)] -
Sqrt[2]*3^(1/4)*ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)] + (3^(1/4)*Log[1 + S
qrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/Sqrt[2] - (3^(1/4)*Log[1 + Sqrt[3
] + 2*x + Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/Sqrt[2]

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Rubi in Sympy [A]  time = 33.6457, size = 162, normalized size = 0.96 \[ 4 \sqrt{2 x + 1} + \frac{\sqrt{2} \sqrt [4]{3} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(2*x + 1) + sqrt(2)*3**(1/4)*log(2*x - sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1
+ sqrt(3))/2 - sqrt(2)*3**(1/4)*log(2*x + sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1 + s
qrt(3))/2 - sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 - 1) - sqrt(2
)*3**(1/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 + 1)

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Mathematica [A]  time = 0.0704926, size = 154, normalized size = 0.92 \[ 4 \sqrt{2 x+1}+\frac{\sqrt [4]{3} \log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}-\frac{\sqrt [4]{3} \log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )-\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*Sqrt[1 + 2*x] + Sqrt[2]*3^(1/4)*ArcTan[1 - Sqrt[2 + 4*x]/3^(1/4)] - Sqrt[2]*3^
(1/4)*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] + (3^(1/4)*Log[3 + Sqrt[3]*(1 + 2*x) - 3
^(3/4)*Sqrt[2 + 4*x]])/Sqrt[2] - (3^(1/4)*Log[3 + Sqrt[3]*(1 + 2*x) + 3^(3/4)*Sq
rt[2 + 4*x]])/Sqrt[2]

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Maple [A]  time = 0.008, size = 120, normalized size = 0.7 \[ 4\,\sqrt{1+2\,x}-\sqrt [4]{3}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2}-{\frac{\sqrt [4]{3}\sqrt{2}}{2}\ln \left ({1 \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) }-\sqrt [4]{3}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(3/2)/(x^2+x+1),x)

[Out]

4*(1+2*x)^(1/2)-3^(1/4)*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))*2^(1/2)-1/2
*3^(1/4)*2^(1/2)*ln((1+2*x+3^(1/2)+3^(1/4)*2^(1/2)*(1+2*x)^(1/2))/(1+2*x+3^(1/2)
-3^(1/4)*2^(1/2)*(1+2*x)^(1/2)))-3^(1/4)*arctan(1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3
/4))*2^(1/2)

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Maxima [A]  time = 0.762007, size = 190, normalized size = 1.13 \[ -3^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 3^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + 4 \, \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3^(1/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x + 1)))
- 3^(1/4)*sqrt(2)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt(2*x + 1
))) - 1/2*3^(1/4)*sqrt(2)*log(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1)
+ 1/2*3^(1/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) +
4*sqrt(2*x + 1)

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Fricas [A]  time = 0.224159, size = 266, normalized size = 1.58 \[ 2 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{3^{\frac{1}{4}} \sqrt{2}}{3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1} + 2 \, \sqrt{2 \, x + 1}}\right ) + 2 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{3^{\frac{1}{4}} \sqrt{2}}{3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1} - 2 \, \sqrt{2 \, x + 1}}\right ) - \frac{1}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + 4 \, \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*3^(1/4)*sqrt(2)*arctan(3^(1/4)*sqrt(2)/(3^(1/4)*sqrt(2) + 2*sqrt(3^(1/4)*sqrt(
2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 2*sqrt(2*x + 1))) + 2*3^(1/4)*sqrt(2)*ar
ctan(-3^(1/4)*sqrt(2)/(3^(1/4)*sqrt(2) - 2*sqrt(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) +
2*x + sqrt(3) + 1) - 2*sqrt(2*x + 1))) - 1/2*3^(1/4)*sqrt(2)*log(3^(1/4)*sqrt(2
)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 1/2*3^(1/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*
sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 4*sqrt(2*x + 1)

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Sympy [A]  time = 13.8114, size = 162, normalized size = 0.96 \[ 4 \sqrt{2 x + 1} + \frac{\sqrt{2} \sqrt [4]{3} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(2*x + 1) + sqrt(2)*3**(1/4)*log(2*x - sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1
+ sqrt(3))/2 - sqrt(2)*3**(1/4)*log(2*x + sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1 + s
qrt(3))/2 - sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 - 1) - sqrt(2
)*3**(1/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 + 1)

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GIAC/XCAS [A]  time = 0.230685, size = 174, normalized size = 1.04 \[ -12^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 12^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{2} \cdot 12^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{2} \cdot 12^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + 4 \, \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x + 1))) - 12^(
1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt(2*x + 1))) - 1/2*12^(
1/4)*ln(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 1/2*12^(1/4)*ln(-3^
(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 4*sqrt(2*x + 1)