3.1310 \(\int \frac{(1+2 x)^{5/2}}{1+x+x^2} \, dx\)

Optimal. Leaf size=170 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.308692, antiderivative size = 170, 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 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*(1 + 2*x)^(3/2))/3 + Sqrt[2]*3^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/
4)] - Sqrt[2]*3^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])/3^(1/4)] - (3^(3/4)*Log
[1 + Sqrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/Sqrt[2] + (3^(3/4)*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 = 34.5328, size = 163, normalized size = 0.96 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0991131, size = 156, normalized size = 0.92 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(5/2)/(x^2+x+1),x)

[Out]

4/3*(1+2*x)^(3/2)-3^(3/4)*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))*2^(1/2)-1
/2*3^(3/4)*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^(3/4)*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 = 0.768606, size = 190, normalized size = 1.12 \[ -3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{2} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{2} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227304, size = 278, normalized size = 1.64 \[ 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{27^{\frac{3}{4}} \sqrt{2}}{27^{\frac{3}{4}} \sqrt{2} + 6 \, \sqrt{27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 18 \, x + 9 \, \sqrt{3} + 9} + 18 \, \sqrt{2 \, x + 1}}\right ) + 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{27^{\frac{3}{4}} \sqrt{2}}{27^{\frac{3}{4}} \sqrt{2} - 2 \, \sqrt{-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81} - 18 \, \sqrt{2 \, x + 1}}\right ) + \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) - \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) + \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*27^(1/4)*sqrt(2)*arctan(27^(3/4)*sqrt(2)/(27^(3/4)*sqrt(2) + 6*sqrt(27^(3/4)*s
qrt(2)*sqrt(2*x + 1) + 18*x + 9*sqrt(3) + 9) + 18*sqrt(2*x + 1))) + 2*27^(1/4)*s
qrt(2)*arctan(-27^(3/4)*sqrt(2)/(27^(3/4)*sqrt(2) - 2*sqrt(-9*27^(3/4)*sqrt(2)*s
qrt(2*x + 1) + 162*x + 81*sqrt(3) + 81) - 18*sqrt(2*x + 1))) + 1/2*27^(1/4)*sqrt
(2)*log(9*27^(3/4)*sqrt(2)*sqrt(2*x + 1) + 162*x + 81*sqrt(3) + 81) - 1/2*27^(1/
4)*sqrt(2)*log(-9*27^(3/4)*sqrt(2)*sqrt(2*x + 1) + 162*x + 81*sqrt(3) + 81) + 4/
3*(2*x + 1)^(3/2)

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Sympy [A]  time = 26.627, size = 163, normalized size = 0.96 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{3} - \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )} + \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )} - 2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - 2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.230681, size = 174, normalized size = 1.02 \[ \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} - 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 108^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{2} \cdot 108^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{2} \cdot 108^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(2*x + 1)^(3/2) - 108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*
sqrt(2*x + 1))) - 108^(1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqr
t(2*x + 1))) + 1/2*108^(1/4)*ln(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) +
1) - 1/2*108^(1/4)*ln(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1)