3.637 \(\int \frac{\sqrt{2+3 x}}{1+x^2} \, dx\)

Optimal. Leaf size=214 \[ \frac{3 \log \left (3 x-\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \log \left (3 x+\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2 \left (2+\sqrt{13}\right )}-2 \sqrt{3 x+2}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{3 x+2}+\sqrt{2 \left (2+\sqrt{13}\right )}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}} \]

[Out]

(-3*ArcTan[(Sqrt[2*(2 + Sqrt[13])] - 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[13])]])/
Sqrt[2*(-2 + Sqrt[13])] + (3*ArcTan[(Sqrt[2*(2 + Sqrt[13])] + 2*Sqrt[2 + 3*x])/S
qrt[2*(-2 + Sqrt[13])]])/Sqrt[2*(-2 + Sqrt[13])] + (3*Log[2 + Sqrt[13] + 3*x - S
qrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])]) - (3*Log[2 + Sq
rt[13] + 3*x + Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])])

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Rubi [A]  time = 0.488285, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{3 \log \left (3 x-\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \log \left (3 x+\sqrt{2 \left (2+\sqrt{13}\right )} \sqrt{3 x+2}+\sqrt{13}+2\right )}{2 \sqrt{2 \left (2+\sqrt{13}\right )}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2 \left (2+\sqrt{13}\right )}-2 \sqrt{3 x+2}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{3 x+2}+\sqrt{2 \left (2+\sqrt{13}\right )}}{\sqrt{2 \left (\sqrt{13}-2\right )}}\right )}{\sqrt{2 \left (\sqrt{13}-2\right )}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + 3*x]/(1 + x^2),x]

[Out]

(-3*ArcTan[(Sqrt[2*(2 + Sqrt[13])] - 2*Sqrt[2 + 3*x])/Sqrt[2*(-2 + Sqrt[13])]])/
Sqrt[2*(-2 + Sqrt[13])] + (3*ArcTan[(Sqrt[2*(2 + Sqrt[13])] + 2*Sqrt[2 + 3*x])/S
qrt[2*(-2 + Sqrt[13])]])/Sqrt[2*(-2 + Sqrt[13])] + (3*Log[2 + Sqrt[13] + 3*x - S
qrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])]) - (3*Log[2 + Sq
rt[13] + 3*x + Sqrt[2*(2 + Sqrt[13])]*Sqrt[2 + 3*x]])/(2*Sqrt[2*(2 + Sqrt[13])])

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Rubi in Sympy [A]  time = 40.0976, size = 223, normalized size = 1.04 \[ \frac{3 \sqrt{2} \log{\left (3 x - \sqrt{2} \sqrt{2 + \sqrt{13}} \sqrt{3 x + 2} + 2 + \sqrt{13} \right )}}{4 \sqrt{2 + \sqrt{13}}} - \frac{3 \sqrt{2} \log{\left (3 x + \sqrt{2} \sqrt{2 + \sqrt{13}} \sqrt{3 x + 2} + 2 + \sqrt{13} \right )}}{4 \sqrt{2 + \sqrt{13}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{3 x + 2} - \frac{\sqrt{4 + 2 \sqrt{13}}}{2}\right )}{\sqrt{-2 + \sqrt{13}}} \right )}}{2 \sqrt{-2 + \sqrt{13}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{3 x + 2} + \frac{\sqrt{4 + 2 \sqrt{13}}}{2}\right )}{\sqrt{-2 + \sqrt{13}}} \right )}}{2 \sqrt{-2 + \sqrt{13}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(2)*log(3*x - sqrt(2)*sqrt(2 + sqrt(13))*sqrt(3*x + 2) + 2 + sqrt(13))/(4*
sqrt(2 + sqrt(13))) - 3*sqrt(2)*log(3*x + sqrt(2)*sqrt(2 + sqrt(13))*sqrt(3*x +
2) + 2 + sqrt(13))/(4*sqrt(2 + sqrt(13))) + 3*sqrt(2)*atan(sqrt(2)*(sqrt(3*x + 2
) - sqrt(4 + 2*sqrt(13))/2)/sqrt(-2 + sqrt(13)))/(2*sqrt(-2 + sqrt(13))) + 3*sqr
t(2)*atan(sqrt(2)*(sqrt(3*x + 2) + sqrt(4 + 2*sqrt(13))/2)/sqrt(-2 + sqrt(13)))/
(2*sqrt(-2 + sqrt(13)))

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Mathematica [C]  time = 0.0793881, size = 59, normalized size = 0.28 \[ \frac{(3-2 i) \tan ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{-2-3 i}}\right )}{\sqrt{-2-3 i}}+\frac{(3+2 i) \tan ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{-2+3 i}}\right )}{\sqrt{-2+3 i}} \]

Antiderivative was successfully verified.

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

[Out]

((3 - 2*I)*ArcTan[Sqrt[2 + 3*x]/Sqrt[-2 - 3*I]])/Sqrt[-2 - 3*I] + ((3 + 2*I)*Arc
Tan[Sqrt[2 + 3*x]/Sqrt[-2 + 3*I]])/Sqrt[-2 + 3*I]

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Maple [B]  time = 0.123, size = 360, normalized size = 1.7 \[ -{\frac{\sqrt{4+2\,\sqrt{13}}}{6}\ln \left ( 2+3\,x+\sqrt{13}-\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }-{\frac{4+2\,\sqrt{13}}{3\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}-\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }+{\frac{\sqrt{4+2\,\sqrt{13}}\sqrt{13}}{12}\ln \left ( 2+3\,x+\sqrt{13}-\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }+{\frac{\sqrt{13} \left ( 4+2\,\sqrt{13} \right ) }{6\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}-\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }+{\frac{\sqrt{4+2\,\sqrt{13}}}{6}\ln \left ( 2+3\,x+\sqrt{13}+\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }-{\frac{4+2\,\sqrt{13}}{3\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}+\sqrt{4+2\,\sqrt{13}} \right ) } \right ) }-{\frac{\sqrt{4+2\,\sqrt{13}}\sqrt{13}}{12}\ln \left ( 2+3\,x+\sqrt{13}+\sqrt{2+3\,x}\sqrt{4+2\,\sqrt{13}} \right ) }+{\frac{\sqrt{13} \left ( 4+2\,\sqrt{13} \right ) }{6\,\sqrt{-4+2\,\sqrt{13}}}\arctan \left ({\frac{1}{\sqrt{-4+2\,\sqrt{13}}} \left ( 2\,\sqrt{2+3\,x}+\sqrt{4+2\,\sqrt{13}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(1/2)/(x^2+1),x)

[Out]

-1/6*(4+2*13^(1/2))^(1/2)*ln(2+3*x+13^(1/2)-(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))-
1/3*(4+2*13^(1/2))/(-4+2*13^(1/2))^(1/2)*arctan((2*(2+3*x)^(1/2)-(4+2*13^(1/2))^
(1/2))/(-4+2*13^(1/2))^(1/2))+1/12*(4+2*13^(1/2))^(1/2)*13^(1/2)*ln(2+3*x+13^(1/
2)-(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))+1/6*13^(1/2)*(4+2*13^(1/2))/(-4+2*13^(1/2
))^(1/2)*arctan((2*(2+3*x)^(1/2)-(4+2*13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2))+1/
6*(4+2*13^(1/2))^(1/2)*ln(2+3*x+13^(1/2)+(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))-1/3
*(4+2*13^(1/2))/(-4+2*13^(1/2))^(1/2)*arctan((2*(2+3*x)^(1/2)+(4+2*13^(1/2))^(1/
2))/(-4+2*13^(1/2))^(1/2))-1/12*(4+2*13^(1/2))^(1/2)*13^(1/2)*ln(2+3*x+13^(1/2)+
(2+3*x)^(1/2)*(4+2*13^(1/2))^(1/2))+1/6*13^(1/2)*(4+2*13^(1/2))/(-4+2*13^(1/2))^
(1/2)*arctan((2*(2+3*x)^(1/2)+(4+2*13^(1/2))^(1/2))/(-4+2*13^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x + 2}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(3*x + 2)/(x^2 + 1), x)

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Fricas [A]  time = 0.2421, size = 743, normalized size = 3.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(13^(1/4)*(sqrt(13) - 2)*log(1/13*(3*13^(1/4)*sqrt(2)*sqrt(3*x + 2)*
(497*sqrt(13) - 1768)*sqrt((2*sqrt(13) - 13)/(4*sqrt(13) - 17)) + 1768*sqrt(13)*
(3*x + 2) + 13*sqrt(13)*(136*sqrt(13) - 497) - 19383*x - 12922)/(136*sqrt(13) -
497)) - 13^(1/4)*(sqrt(13) - 2)*log(-1/13*(3*13^(1/4)*sqrt(2)*sqrt(3*x + 2)*(497
*sqrt(13) - 1768)*sqrt((2*sqrt(13) - 13)/(4*sqrt(13) - 17)) - 1768*sqrt(13)*(3*x
+ 2) - 13*sqrt(13)*(136*sqrt(13) - 497) + 19383*x + 12922)/(136*sqrt(13) - 497)
) - 12*13^(1/4)*arctan(13^(1/4)*(sqrt(13) - 2)/(sqrt(2)*sqrt(1/13)*(sqrt(13) - 2
)*sqrt((3*13^(1/4)*sqrt(2)*sqrt(3*x + 2)*(497*sqrt(13) - 1768)*sqrt((2*sqrt(13)
- 13)/(4*sqrt(13) - 17)) + 1768*sqrt(13)*(3*x + 2) + 13*sqrt(13)*(136*sqrt(13) -
497) - 19383*x - 12922)/(136*sqrt(13) - 497))*sqrt((2*sqrt(13) - 13)/(4*sqrt(13
) - 17)) + sqrt(2)*sqrt(3*x + 2)*(sqrt(13) - 2)*sqrt((2*sqrt(13) - 13)/(4*sqrt(1
3) - 17)) - 3*13^(1/4))) - 12*13^(1/4)*arctan(13^(1/4)*(sqrt(13) - 2)/(sqrt(2)*s
qrt(1/13)*(sqrt(13) - 2)*sqrt(-(3*13^(1/4)*sqrt(2)*sqrt(3*x + 2)*(497*sqrt(13) -
1768)*sqrt((2*sqrt(13) - 13)/(4*sqrt(13) - 17)) - 1768*sqrt(13)*(3*x + 2) - 13*
sqrt(13)*(136*sqrt(13) - 497) + 19383*x + 12922)/(136*sqrt(13) - 497))*sqrt((2*s
qrt(13) - 13)/(4*sqrt(13) - 17)) + sqrt(2)*sqrt(3*x + 2)*(sqrt(13) - 2)*sqrt((2*
sqrt(13) - 13)/(4*sqrt(13) - 17)) + 3*13^(1/4))))/((sqrt(13) - 2)*sqrt((2*sqrt(1
3) - 13)/(4*sqrt(13) - 17)))

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Sympy [A]  time = 9.14882, size = 32, normalized size = 0.15 \[ 6 \operatorname{RootSum}{\left (20736 t^{4} + 576 t^{2} + 13, \left ( t \mapsto t \log{\left (576 t^{3} + 8 t + \sqrt{3 x + 2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*RootSum(20736*_t**4 + 576*_t**2 + 13, Lambda(_t, _t*log(576*_t**3 + 8*_t + sqr
t(3*x + 2))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x + 2}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(3*x + 2)/(x^2 + 1), x)