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

Optimal. Leaf size=80 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2}-2 \sqrt{x+1}+(1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1-i}}\right )+(1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{1+i}}\right ) \]

[Out]

-2*Sqrt[1 + x] - (2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5 + (1 - I)^(3/2)*ArcTa
nh[Sqrt[1 + x]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + x]/Sqrt[1 + I]]

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Rubi [B]  time = 0.566475, antiderivative size = 224, normalized size of antiderivative = 2.8, number of steps used = 16, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{2}{5} (x+1)^{5/2}-\frac{2}{3} (x+1)^{3/2}-2 \sqrt{x+1}-\frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}}}+\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}}}-\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.6651, size = 211, normalized size = 2.64 \[ \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{5} - \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{x + 1} - \frac{\log{\left (x - \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{2 \sqrt{1 + \sqrt{2}}} + \frac{\log{\left (x + \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{2 \sqrt{1 + \sqrt{2}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} - \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{\sqrt{-1 + \sqrt{2}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} + \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{\sqrt{-1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)**(5/2)/5 - 2*(x + 1)**(3/2)/3 - 2*sqrt(x + 1) - log(x - sqrt(2)*sqrt(1
+ sqrt(2))*sqrt(x + 1) + 1 + sqrt(2))/(2*sqrt(1 + sqrt(2))) + log(x + sqrt(2)*s
qrt(1 + sqrt(2))*sqrt(x + 1) + 1 + sqrt(2))/(2*sqrt(1 + sqrt(2))) + atan(sqrt(2)
*(sqrt(x + 1) - sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2)))/sqrt(-1 + sqrt(2)) +
atan(sqrt(2)*(sqrt(x + 1) + sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2)))/sqrt(-1 +
sqrt(2))

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Mathematica [A]  time = 0.097183, size = 70, normalized size = 0.88 \[ \frac{2}{15} \sqrt{x+1} \left (3 x^2+x-17\right )-(-1-i)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1-i}}\right )-(-1+i)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1+i}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 + x]*(-17 + x + 3*x^2))/15 - (-1 - I)^(3/2)*ArcTan[Sqrt[1 + x]/Sqrt[-1
- I]] - (-1 + I)^(3/2)*ArcTan[Sqrt[1 + x]/Sqrt[-1 + I]]

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Maple [B]  time = 0.046, size = 443, normalized size = 5.5 \[{\frac{2}{5} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{2}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+x}+{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}}{2}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}}{2}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(1+x)^(5/2)-2/3*(1+x)^(3/2)-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)*2^(1/2)-1/2*(2+2*2^(1/2))^(1/2)*ln(1+x+2^(
1/2)-(1+x)^(1/2)*(2+2*2^(1/2))^(1/2))+1/2/(-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+2*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))*(2
+2*2^(1/2))+2/(-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)-1/4*ln(1+x+2^(1/2)+(1+x)^(1/2)*(2+2*2^(1/2))^(1/2))*
(2+2*2^(1/2))^(1/2)*2^(1/2)+1/2*(2+2*2^(1/2))^(1/2)*ln(1+x+2^(1/2)+(1+x)^(1/2)*(
2+2*2^(1/2))^(1/2))+1/2/(-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+2*2^(1/2))*2^(1/2)-1/(-2+2*2^(1/2))^(1/2)*arcta
n((2*(1+x)^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(2+2*2^(1/2))+2/(-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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^3 + 1)*sqrt(x + 1)/(x^2 + 1), x)

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Fricas [A]  time = 0.285619, size = 582, normalized size = 7.28 \[ -\frac{\sqrt{2}{\left (4 \, \sqrt{2}{\left (3 \, x^{2} - \sqrt{2}{\left (3 \, x^{2} + x - 17\right )} + x - 17\right )} \sqrt{x + 1} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 15 \cdot 8^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (2 \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 4 \, x + 4 \, \sqrt{2} + 4\right ) + 15 \cdot 8^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (-2 \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 4 \, x + 4 \, \sqrt{2} + 4\right ) - 60 \cdot 8^{\frac{1}{4}} \arctan \left (\frac{8^{\frac{1}{4}}{\left (\sqrt{2} - 2\right )}}{\sqrt{2} \sqrt{2 \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 4 \, x + 4 \, \sqrt{2} + 4}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 2 \, \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 8^{\frac{1}{4}} \sqrt{2}}\right ) - 60 \cdot 8^{\frac{1}{4}} \arctan \left (\frac{8^{\frac{1}{4}}{\left (\sqrt{2} - 2\right )}}{\sqrt{2} \sqrt{-2 \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{x + 1} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 4 \, x + 4 \, \sqrt{2} + 4}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 2 \, \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 8^{\frac{1}{4}} \sqrt{2}}\right )\right )}}{60 \,{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*sqrt(2)*(4*sqrt(2)*(3*x^2 - sqrt(2)*(3*x^2 + x - 17) + x - 17)*sqrt(x + 1)
*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 15*8^(1/4)*(sqrt(2) - 1)*log(2*8^(1/4)*sq
rt(2)*sqrt(x + 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 4*x + 4*sqrt(2) + 4) + 1
5*8^(1/4)*(sqrt(2) - 1)*log(-2*8^(1/4)*sqrt(2)*sqrt(x + 1)*sqrt((sqrt(2) - 2)/(2
*sqrt(2) - 3)) + 4*x + 4*sqrt(2) + 4) - 60*8^(1/4)*arctan(8^(1/4)*(sqrt(2) - 2)/
(sqrt(2)*sqrt(2*8^(1/4)*sqrt(2)*sqrt(x + 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3))
+ 4*x + 4*sqrt(2) + 4)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2*sqr
t(2)*sqrt(x + 1)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 8^(1/4)*sqr
t(2))) - 60*8^(1/4)*arctan(8^(1/4)*(sqrt(2) - 2)/(sqrt(2)*sqrt(-2*8^(1/4)*sqrt(2
)*sqrt(x + 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 4*x + 4*sqrt(2) + 4)*(sqrt(2
) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2*sqrt(2)*sqrt(x + 1)*(sqrt(2) - 1)
*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 8^(1/4)*sqrt(2))))/((sqrt(2) - 1)*sqrt((s
qrt(2) - 2)/(2*sqrt(2) - 3)))

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Sympy [A]  time = 14.3451, size = 56, normalized size = 0.7 \[ \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{5} - \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{x + 1} + 4 \operatorname{RootSum}{\left (512 t^{4} + 32 t^{2} + 1, \left ( t \mapsto t \log{\left (- 128 t^{3} + \sqrt{x + 1} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)**(5/2)/5 - 2*(x + 1)**(3/2)/3 - 2*sqrt(x + 1) + 4*RootSum(512*_t**4 +
32*_t**2 + 1, Lambda(_t, _t*log(-128*_t**3 + sqrt(x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^3 + 1)*sqrt(x + 1)/(x^2 + 1), x)