3.315 \(\int \frac{x \sqrt{1+x}}{1+x^2} \, dx\)

Optimal. Leaf size=214 \[ 2 \sqrt{x+1}+\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )-\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{\sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{\sqrt{2 \left (1+\sqrt{2}\right )}} \]

[Out]

2*Sqrt[1 + x] + ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-1 + Sqrt
[2])]]/Sqrt[2*(1 + Sqrt[2])] - ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + x])/Sq
rt[2*(-1 + Sqrt[2])]]/Sqrt[2*(1 + Sqrt[2])] + (Sqrt[(1 + Sqrt[2])/2]*Log[1 + Sqr
t[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + x]])/2 - (Sqrt[(1 + Sqrt[2])/2]*Log[1
+ Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + x]])/2

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Rubi [A]  time = 0.526942, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ 2 \sqrt{x+1}+\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )-\frac{1}{2} \sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{\sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{\sqrt{2 \left (1+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[1 + x] + ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-1 + Sqrt
[2])]]/Sqrt[2*(1 + Sqrt[2])] - ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + x])/Sq
rt[2*(-1 + Sqrt[2])]]/Sqrt[2*(1 + Sqrt[2])] + (Sqrt[(1 + Sqrt[2])/2]*Log[1 + Sqr
t[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + x]])/2 - (Sqrt[(1 + Sqrt[2])/2]*Log[1
+ Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + x]])/2

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Rubi in Sympy [A]  time = 51.1913, size = 323, normalized size = 1.51 \[ 2 \sqrt{x + 1} + \frac{\left (\frac{\sqrt{2}}{2} + 1\right ) \log{\left (x - \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{2 \sqrt{1 + \sqrt{2}}} - \frac{\left (\frac{\sqrt{2}}{2} + 1\right ) \log{\left (x + \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{2 \sqrt{1 + \sqrt{2}}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2} \sqrt{1 + \sqrt{2}} \left (\sqrt{2} + 2\right )}{2} + 2 \sqrt{2} \sqrt{1 + \sqrt{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} - \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}} \sqrt{1 + \sqrt{2}}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2} \sqrt{1 + \sqrt{2}} \left (\sqrt{2} + 2\right )}{2} + 2 \sqrt{2} \sqrt{1 + \sqrt{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} + \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}} \sqrt{1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) + (sqrt(2)/2 + 1)*log(x - sqrt(2)*sqrt(1 + sqrt(2))*sqrt(x + 1) +
1 + sqrt(2))/(2*sqrt(1 + sqrt(2))) - (sqrt(2)/2 + 1)*log(x + sqrt(2)*sqrt(1 + sq
rt(2))*sqrt(x + 1) + 1 + sqrt(2))/(2*sqrt(1 + sqrt(2))) - sqrt(2)*(-sqrt(2)*sqrt
(1 + sqrt(2))*(sqrt(2) + 2)/2 + 2*sqrt(2)*sqrt(1 + sqrt(2)))*atan(sqrt(2)*(sqrt(
x + 1) - sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2)))/(2*sqrt(-1 + sqrt(2))*sqrt(1
+ sqrt(2))) - sqrt(2)*(-sqrt(2)*sqrt(1 + sqrt(2))*(sqrt(2) + 2)/2 + 2*sqrt(2)*s
qrt(1 + sqrt(2)))*atan(sqrt(2)*(sqrt(x + 1) + sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + s
qrt(2)))/(2*sqrt(-1 + sqrt(2))*sqrt(1 + sqrt(2)))

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Mathematica [C]  time = 0.0304022, size = 60, normalized size = 0.28 \[ 2 \sqrt{x+1}-\sqrt{1-i} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1-i}}\right )-\sqrt{1+i} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1+i}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[1 + x] - Sqrt[1 - I]*ArcTanh[Sqrt[1 + x]/Sqrt[1 - I]] - Sqrt[1 + I]*ArcTa
nh[Sqrt[1 + x]/Sqrt[1 + I]]

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Maple [A]  time = 0.121, size = 240, normalized size = 1.1 \[ 2\,\sqrt{1+x}+{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+x)^(1/2)/(x^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1} x}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + 1)*x/(x^2 + 1), x)

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Fricas [A]  time = 0.304006, size = 711, normalized size = 3.32 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} + 1\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} - 2^{\frac{1}{4}}{\left (\sqrt{2} + 1\right )} \log \left (\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (41 \, \sqrt{2} + 58\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} + 24 \, \sqrt{2}{\left (x + 1\right )} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} + 17\right )} + 34 \, x + 34}{2 \,{\left (12 \, \sqrt{2} + 17\right )}}\right ) + 2^{\frac{1}{4}}{\left (\sqrt{2} + 1\right )} \log \left (-\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (41 \, \sqrt{2} + 58\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} - 24 \, \sqrt{2}{\left (x + 1\right )} - 2 \, \sqrt{2}{\left (12 \, \sqrt{2} + 17\right )} - 34 \, x - 34}{2 \,{\left (12 \, \sqrt{2} + 17\right )}}\right ) + 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} + 1\right )} \sqrt{\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (41 \, \sqrt{2} + 58\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} + 24 \, \sqrt{2}{\left (x + 1\right )} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} + 17\right )} + 34 \, x + 34}{12 \, \sqrt{2} + 17}} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} + \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} + 1\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} + 2^{\frac{1}{4}}{\left (\sqrt{2} + 1\right )}}\right ) + 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} + 1\right )} \sqrt{-\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (41 \, \sqrt{2} + 58\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} - 24 \, \sqrt{2}{\left (x + 1\right )} - 2 \, \sqrt{2}{\left (12 \, \sqrt{2} + 17\right )} - 34 \, x - 34}{12 \, \sqrt{2} + 17}} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} + \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} + 1\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}} - 2^{\frac{1}{4}}{\left (\sqrt{2} + 1\right )}}\right )\right )}}{4 \,{\left (\sqrt{2} + 1\right )} \sqrt{\frac{\sqrt{2} + 2}{2 \, \sqrt{2} + 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(4*sqrt(2)*sqrt(x + 1)*(sqrt(2) + 1)*sqrt((sqrt(2) + 2)/(2*sqrt(2) +
3)) - 2^(1/4)*(sqrt(2) + 1)*log(1/2*(2^(3/4)*sqrt(x + 1)*(41*sqrt(2) + 58)*sqrt
((sqrt(2) + 2)/(2*sqrt(2) + 3)) + 24*sqrt(2)*(x + 1) + 2*sqrt(2)*(12*sqrt(2) + 1
7) + 34*x + 34)/(12*sqrt(2) + 17)) + 2^(1/4)*(sqrt(2) + 1)*log(-1/2*(2^(3/4)*sqr
t(x + 1)*(41*sqrt(2) + 58)*sqrt((sqrt(2) + 2)/(2*sqrt(2) + 3)) - 24*sqrt(2)*(x +
1) - 2*sqrt(2)*(12*sqrt(2) + 17) - 34*x - 34)/(12*sqrt(2) + 17)) + 4*2^(1/4)*ar
ctan(2^(1/4)/(sqrt(2)*sqrt(1/2)*(sqrt(2) + 1)*sqrt((2^(3/4)*sqrt(x + 1)*(41*sqrt
(2) + 58)*sqrt((sqrt(2) + 2)/(2*sqrt(2) + 3)) + 24*sqrt(2)*(x + 1) + 2*sqrt(2)*(
12*sqrt(2) + 17) + 34*x + 34)/(12*sqrt(2) + 17))*sqrt((sqrt(2) + 2)/(2*sqrt(2) +
3)) + sqrt(2)*sqrt(x + 1)*(sqrt(2) + 1)*sqrt((sqrt(2) + 2)/(2*sqrt(2) + 3)) + 2
^(1/4)*(sqrt(2) + 1))) + 4*2^(1/4)*arctan(2^(1/4)/(sqrt(2)*sqrt(1/2)*(sqrt(2) +
1)*sqrt(-(2^(3/4)*sqrt(x + 1)*(41*sqrt(2) + 58)*sqrt((sqrt(2) + 2)/(2*sqrt(2) +
3)) - 24*sqrt(2)*(x + 1) - 2*sqrt(2)*(12*sqrt(2) + 17) - 34*x - 34)/(12*sqrt(2)
+ 17))*sqrt((sqrt(2) + 2)/(2*sqrt(2) + 3)) + sqrt(2)*sqrt(x + 1)*(sqrt(2) + 1)*s
qrt((sqrt(2) + 2)/(2*sqrt(2) + 3)) - 2^(1/4)*(sqrt(2) + 1))))/((sqrt(2) + 1)*sqr
t((sqrt(2) + 2)/(2*sqrt(2) + 3)))

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Sympy [A]  time = 11.3025, size = 68, normalized size = 0.32 \[ 2 \sqrt{x + 1} - 4 \operatorname{RootSum}{\left (512 t^{4} + 32 t^{2} + 1, \left ( t \mapsto t \log{\left (- 128 t^{3} + \sqrt{x + 1} \right )} \right )\right )} + 2 \operatorname{RootSum}{\left (128 t^{4} + 16 t^{2} + 1, \left ( t \mapsto t \log{\left (64 t^{3} + 4 t + \sqrt{x + 1} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) - 4*RootSum(512*_t**4 + 32*_t**2 + 1, Lambda(_t, _t*log(-128*_t**3
+ sqrt(x + 1)))) + 2*RootSum(128*_t**4 + 16*_t**2 + 1, Lambda(_t, _t*log(64*_t*
*3 + 4*_t + sqrt(x + 1))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1} x}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + 1)*x/(x^2 + 1), x)