3.1314 \(\int \frac{1}{(1+2 x)^{3/2} \left (1+x+x^2\right )} \, dx\)

Optimal. Leaf size=180 \[ -\frac{4}{3 \sqrt{2 x+1}}-\frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{3 \sqrt [4]{3}} \]

[Out]

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

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Rubi [A]  time = 0.309213, antiderivative size = 180, 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 \[ -\frac{4}{3 \sqrt{2 x+1}}-\frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{3 \sqrt{2} \sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{3 \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{3 \sqrt [4]{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.5433, size = 167, normalized size = 0.93 \[ - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{18} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{18} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )}}{9} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )}}{9} - \frac{4}{3 \sqrt{2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*3**(3/4)*log(2*x - sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/18 + s
qrt(2)*3**(3/4)*log(2*x + sqrt(2)*3**(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/18 - sqr
t(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 - 1)/9 - sqrt(2)*3**(3/4)*at
an(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3 + 1)/9 - 4/(3*sqrt(2*x + 1))

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Mathematica [A]  time = 0.179049, size = 159, normalized size = 0.88 \[ \frac{1}{18} \left (-\frac{24}{\sqrt{2 x+1}}-\sqrt{2} 3^{3/4} \log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )+\sqrt{2} 3^{3/4} \log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )+2 \sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )-2 \sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-24/Sqrt[1 + 2*x] + 2*Sqrt[2]*3^(3/4)*ArcTan[1 - Sqrt[2 + 4*x]/3^(1/4)] - 2*Sqr
t[2]*3^(3/4)*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] - Sqrt[2]*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]])/18

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Maple [A]  time = 0.011, size = 120, normalized size = 0.7 \[ -{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{9}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }-{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{9}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }-{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{18}\ln \left ({1 \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) }-{\frac{4}{3}{\frac{1}{\sqrt{1+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*3^(3/4)*arctan(1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))*2^(1/2)-1/9*3^(3/4)*arc
tan(-1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))*2^(1/2)-1/18*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)))-4/3/(1+2*x)^(1/2)

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Maxima [A]  time = 0.762848, size = 190, normalized size = 1.06 \[ -\frac{1}{9} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{9} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{18} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{18} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{4}{3 \, \sqrt{2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x +
1))) - 1/9*3^(3/4)*sqrt(2)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt
(2*x + 1))) + 1/18*3^(3/4)*sqrt(2)*log(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqr
t(3) + 1) - 1/18*3^(3/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt
(3) + 1) - 4/3/sqrt(2*x + 1)

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Fricas [A]  time = 0.228665, size = 290, normalized size = 1.61 \[ \frac{3^{\frac{3}{4}} \sqrt{2}{\left (4 \, \sqrt{2 \, x + 1} \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + \sqrt{6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18} + 3}\right ) + 4 \, \sqrt{2 \, x + 1} \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + \sqrt{-6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18} - 3}\right ) + \sqrt{2 \, x + 1} \log \left (6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18\right ) - \sqrt{2 \, x + 1} \log \left (-6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18\right ) - 4 \cdot 3^{\frac{1}{4}} \sqrt{2}\right )}}{18 \, \sqrt{2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*3^(3/4)*sqrt(2)*(4*sqrt(2*x + 1)*arctan(3/(3^(3/4)*sqrt(2)*sqrt(2*x + 1) +
sqrt(6*3^(3/4)*sqrt(2)*sqrt(2*x + 1) + 6*sqrt(3)*(2*x + 1) + 18) + 3)) + 4*sqrt(
2*x + 1)*arctan(3/(3^(3/4)*sqrt(2)*sqrt(2*x + 1) + sqrt(-6*3^(3/4)*sqrt(2)*sqrt(
2*x + 1) + 6*sqrt(3)*(2*x + 1) + 18) - 3)) + sqrt(2*x + 1)*log(6*3^(3/4)*sqrt(2)
*sqrt(2*x + 1) + 6*sqrt(3)*(2*x + 1) + 18) - sqrt(2*x + 1)*log(-6*3^(3/4)*sqrt(2
)*sqrt(2*x + 1) + 6*sqrt(3)*(2*x + 1) + 18) - 4*3^(1/4)*sqrt(2))/sqrt(2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (2 x + 1\right )^{\frac{3}{2}} \left (x^{2} + x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((2*x + 1)**(3/2)*(x**2 + x + 1)), x)

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GIAC/XCAS [A]  time = 0.226783, size = 174, normalized size = 0.97 \[ -\frac{1}{9} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{9} \cdot 108^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{18} \cdot 108^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{18} \cdot 108^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{4}{3 \, \sqrt{2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2*sqrt(2*x + 1))) -
1/9*108^(1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*sqrt(2*x + 1)))
+ 1/18*108^(1/4)*ln(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) - 1/18*10
8^(1/4)*ln(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) - 4/3/sqrt(2*x +
1)