3.1309 \(\int \frac{(1+2 x)^{7/2}}{1+x+x^2} \, dx\)

Optimal. Leaf size=183 \[ \frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

[Out]

-12*Sqrt[1 + 2*x] + (4*(1 + 2*x)^(5/2))/5 - 3*Sqrt[2]*3^(1/4)*ArcTan[1 - (Sqrt[2
]*Sqrt[1 + 2*x])/3^(1/4)] + 3*Sqrt[2]*3^(1/4)*ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])
/3^(1/4)] - (3*3^(1/4)*Log[1 + Sqrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/S
qrt[2] + (3*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 = 0.379684, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)^(7/2)/(1 + x + x^2),x]

[Out]

-12*Sqrt[1 + 2*x] + (4*(1 + 2*x)^(5/2))/5 - 3*Sqrt[2]*3^(1/4)*ArcTan[1 - (Sqrt[2
]*Sqrt[1 + 2*x])/3^(1/4)] + 3*Sqrt[2]*3^(1/4)*ArcTan[1 + (Sqrt[2]*Sqrt[1 + 2*x])
/3^(1/4)] - (3*3^(1/4)*Log[1 + Sqrt[3] + 2*x - Sqrt[2]*3^(1/4)*Sqrt[1 + 2*x]])/S
qrt[2] + (3*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 = 40.0964, size = 180, normalized size = 0.98 \[ \frac{4 \left (2 x + 1\right )^{\frac{5}{2}}}{5} - 12 \sqrt{2 x + 1} - \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + 3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} + 3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(7/2)/(x**2+x+1),x)

[Out]

4*(2*x + 1)**(5/2)/5 - 12*sqrt(2*x + 1) - 3*sqrt(2)*3**(1/4)*log(2*x - sqrt(2)*3
**(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/2 + 3*sqrt(2)*3**(1/4)*log(2*x + sqrt(2)*3*
*(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/2 + 3*sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)
*sqrt(2*x + 1)/3 - 1) + 3*sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*sqrt(2*x + 1)/3
+ 1)

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Mathematica [A]  time = 0.17964, size = 169, normalized size = 0.92 \[ \frac{4}{5} (2 x+1)^{5/2}-12 \sqrt{2 x+1}-\frac{3 \sqrt [4]{3} \log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\frac{3 \sqrt [4]{3} \log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}-3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )+3 \sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)^(7/2)/(1 + x + x^2),x]

[Out]

-12*Sqrt[1 + 2*x] + (4*(1 + 2*x)^(5/2))/5 - 3*Sqrt[2]*3^(1/4)*ArcTan[1 - Sqrt[2
+ 4*x]/3^(1/4)] + 3*Sqrt[2]*3^(1/4)*ArcTan[1 + Sqrt[2 + 4*x]/3^(1/4)] - (3*3^(1/
4)*Log[3 + Sqrt[3]*(1 + 2*x) - 3^(3/4)*Sqrt[2 + 4*x]])/Sqrt[2] + (3*3^(1/4)*Log[
3 + Sqrt[3]*(1 + 2*x) + 3^(3/4)*Sqrt[2 + 4*x]])/Sqrt[2]

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Maple [A]  time = 0.015, size = 129, normalized size = 0.7 \[{\frac{4}{5} \left ( 1+2\,x \right ) ^{{\frac{5}{2}}}}-12\,\sqrt{1+2\,x}+3\,\sqrt [4]{3}\arctan \left ( 1+1/3\,\sqrt{2}\sqrt{1+2\,x}{3}^{3/4} \right ) \sqrt{2}+3\,\sqrt [4]{3}\arctan \left ( -1+1/3\,\sqrt{2}\sqrt{1+2\,x}{3}^{3/4} \right ) \sqrt{2}+{\frac{3\,\sqrt [4]{3}\sqrt{2}}{2}\ln \left ({1 \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(7/2)/(x^2+x+1),x)

[Out]

4/5*(1+2*x)^(5/2)-12*(1+2*x)^(1/2)+3*3^(1/4)*arctan(1+1/3*2^(1/2)*(1+2*x)^(1/2)*
3^(3/4))*2^(1/2)+3*3^(1/4)*arctan(-1+1/3*2^(1/2)*(1+2*x)^(1/2)*3^(3/4))*2^(1/2)+
3/2*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)))

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Maxima [A]  time = 0.760116, size = 203, normalized size = 1.11 \[ \frac{4}{5} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} + 3 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + 3 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{3}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{3}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - 12 \, \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(x^2 + x + 1),x, algorithm="maxima")

[Out]

4/5*(2*x + 1)^(5/2) + 3*3^(1/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt
(2) + 2*sqrt(2*x + 1))) + 3*3^(1/4)*sqrt(2)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)
*sqrt(2) - 2*sqrt(2*x + 1))) + 3/2*3^(1/4)*sqrt(2)*log(3^(1/4)*sqrt(2)*sqrt(2*x
+ 1) + 2*x + sqrt(3) + 1) - 3/2*3^(1/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)*sqrt(2*x +
1) + 2*x + sqrt(3) + 1) - 12*sqrt(2*x + 1)

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Fricas [A]  time = 0.227063, size = 279, normalized size = 1.52 \[ -6 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{3^{\frac{1}{4}} \sqrt{2}}{3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1} + 2 \, \sqrt{2 \, x + 1}}\right ) - 6 \cdot 3^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{3^{\frac{1}{4}} \sqrt{2}}{3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1} - 2 \, \sqrt{2 \, x + 1}}\right ) + \frac{3}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{3}{2} \cdot 3^{\frac{1}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{8}{5} \,{\left (2 \, x^{2} + 2 \, x - 7\right )} \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(x^2 + x + 1),x, algorithm="fricas")

[Out]

-6*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))) - 6*3^(1/4)*sqrt(2)*a
rctan(-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))) + 3/2*3^(1/4)*sqrt(2)*log(3^(1/4)*sqrt(
2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) - 3/2*3^(1/4)*sqrt(2)*log(-3^(1/4)*sqrt(2)
*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) + 8/5*(2*x^2 + 2*x - 7)*sqrt(2*x + 1)

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Sympy [A]  time = 57.5205, size = 180, normalized size = 0.98 \[ \frac{4 \left (2 x + 1\right )^{\frac{5}{2}}}{5} - 12 \sqrt{2 x + 1} - \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + 3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} + 3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x)**(7/2)/(x**2+x+1),x)

[Out]

4*(2*x + 1)**(5/2)/5 - 12*sqrt(2*x + 1) - 3*sqrt(2)*3**(1/4)*log(2*x - sqrt(2)*3
**(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/2 + 3*sqrt(2)*3**(1/4)*log(2*x + sqrt(2)*3*
*(1/4)*sqrt(2*x + 1) + 1 + sqrt(3))/2 + 3*sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)
*sqrt(2*x + 1)/3 - 1) + 3*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 = 0.232513, size = 186, normalized size = 1.02 \[ \frac{4}{5} \,{\left (2 \, x + 1\right )}^{\frac{5}{2}} + 3 \cdot 12^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + 3 \cdot 12^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{3}{2} \cdot 12^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{3}{2} \cdot 12^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - 12 \, \sqrt{2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(x^2 + x + 1),x, algorithm="giac")

[Out]

4/5*(2*x + 1)^(5/2) + 3*12^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) + 2
*sqrt(2*x + 1))) + 3*12^(1/4)*arctan(-1/6*3^(3/4)*sqrt(2)*(3^(1/4)*sqrt(2) - 2*s
qrt(2*x + 1))) + 3/2*12^(1/4)*ln(3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) +
1) - 3/2*12^(1/4)*ln(-3^(1/4)*sqrt(2)*sqrt(2*x + 1) + 2*x + sqrt(3) + 1) - 12*s
qrt(2*x + 1)