\(\int \frac {\sqrt {1+x} (1+x^3)}{1+x^2} \, dx\) [129]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=-2 \sqrt {1+x}-\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2}+(1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {1-i}}\right )+(1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {1+i}}\right ) \] Output:

-2*(1+x)^(1/2)-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)+(1-I)^(3/2)*arctanh((1+x)^( 
1/2)/(1-I)^(1/2))+(1+I)^(3/2)*arctanh((1+x)^(1/2)/(1+I)^(1/2))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\frac {2}{15} \sqrt {1+x} \left (-17+x+3 x^2\right )+\sqrt {2+2 i} \arctan \left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {1+x}\right )+\sqrt {2-2 i} \arctan \left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {1+x}\right ) \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(224\) vs. \(2(80)=160\).

Time = 0.90 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.80, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2156, 2160, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2156

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

\(\Big \downarrow \) 2160

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

\(\Big \downarrow \) 2009

\(\displaystyle -\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2}-2 \sqrt {x+1}-\frac {\log \left (x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}\)

Input:

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

Output:

-2*Sqrt[1 + x] - (2*(1 + x)^(3/2))/3 + (2*(1 + x)^(5/2))/5 - Sqrt[1 + Sqrt 
[2]]*ArcTan[(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])]*Sq 
rt[1 + x]]/(2*Sqrt[1 + Sqrt[2]])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2156

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[(d + e*x)^(m + 1)*PolynomialQuotient[Pq, d + e*x, x]*(a + b*x^2)^p, 
x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemaind 
er[Pq, d + e*x, x], 0]
 

rule 2160

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, 
 d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(58)=116\).

Time = 0.62 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.61

method	result	size
derivativedivides	\(\frac {2 \left (1+x \right )^{\frac {5}{2}}}{5}-\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-2 \sqrt {1+x}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x -\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\)	\(289\)
default	\(\frac {2 \left (1+x \right )^{\frac {5}{2}}}{5}-\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-2 \sqrt {1+x}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x -\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right )}{4}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\)	\(289\)
trager	\(\left (\frac {2}{5} x^{2}+\frac {2}{15} x -\frac {34}{15}\right ) \sqrt {1+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right ) \ln \left (\frac {1024 \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right ) \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{4} x +192 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right ) x +160 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right )+192 \sqrt {1+x}\, \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right ) x +15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+16 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}+1\right )-2 \sqrt {1+x}}{32 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2} x +x -1}\right )+4 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2048 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{5} x -128 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{3} x +96 \sqrt {1+x}\, \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2}-320 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{3}+2 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right ) x +7 \sqrt {1+x}+10 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )}{32 \operatorname {RootOf}\left (512 \textit {\_Z}^{4}+32 \textit {\_Z}^{2}+1\right )^{2} x +x +1}\right )\)	\(426\)
risch	\(\frac {2 \left (3 x^{2}+x -17\right ) \sqrt {1+x}}{15}-\frac {\ln \left (1+x +\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}\, \sqrt {2}}{4}+\frac {\ln \left (1+x +\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}{2}+\frac {\arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \left (2+2 \sqrt {2}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \left (2+2 \sqrt {2}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}+\frac {\ln \left (1+x -\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}\, \sqrt {2}}{4}-\frac {\ln \left (1+x -\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}+\sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}{2}+\frac {\arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \left (2+2 \sqrt {2}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \left (2+2 \sqrt {2}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}\)	\(437\)

Input:

int((1+x)^(1/2)*(x^3+1)/(x^2+1),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (50) = 100\).

Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\frac {2}{15} \, {\left (3 \, x^{2} + x - 17\right )} \sqrt {x + 1} + \sqrt {\sqrt {2} + 1} \arctan \left ({\left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {2} \sqrt {x + 1}\right )} \sqrt {\sqrt {2} + 1}\right ) - \sqrt {\sqrt {2} + 1} \arctan \left ({\left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} - \sqrt {2} \sqrt {x + 1}\right )} \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} + x + \sqrt {2} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-\sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {\sqrt {2} - 1} + x + \sqrt {2} + 1\right ) \] Input:

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

Output:

2/15*(3*x^2 + x - 17)*sqrt(x + 1) + sqrt(sqrt(2) + 1)*arctan(((sqrt(2) + 1 
)*sqrt(sqrt(2) - 1) + sqrt(2)*sqrt(x + 1))*sqrt(sqrt(2) + 1)) - sqrt(sqrt( 
2) + 1)*arctan(((sqrt(2) + 1)*sqrt(sqrt(2) - 1) - sqrt(2)*sqrt(x + 1))*sqr 
t(sqrt(2) + 1)) + 1/2*sqrt(sqrt(2) - 1)*log(sqrt(x + 1)*(sqrt(2) + 2)*sqrt 
(sqrt(2) - 1) + x + sqrt(2) + 1) - 1/2*sqrt(sqrt(2) - 1)*log(-sqrt(x + 1)* 
(sqrt(2) + 2)*sqrt(sqrt(2) - 1) + x + sqrt(2) + 1)
 

Sympy [F]

\[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )}{x^{2} + 1}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\int { \frac {{\left (x^{3} + 1\right )} \sqrt {x + 1}}{x^{2} + 1} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (50) = 100\).

Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} + \sqrt {\sqrt {2} + 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) - 2 \, \sqrt {x + 1} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.19 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=\frac {2\,{\left (x+1\right )}^{5/2}}{5}-\frac {2\,{\left (x+1\right )}^{3/2}}{3}-2\,\sqrt {x+1}-\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {x+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right ) \] Input:

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

Output:

(2*(x + 1)^(5/2))/5 - (2*(x + 1)^(3/2))/3 - 2*(x + 1)^(1/2) - atan((2^(1/2 
)*(- 2^(1/2)/4 - 1/4)^(1/2)*(x + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2 
)*(- 2^(1/2)/4 - 1/4)^(1/2) - 64) - (2^(1/2)*(2^(1/2)/4 - 1/4)^(1/2)*(x + 
1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) - 64) 
)*((- 2^(1/2)/4 - 1/4)^(1/2)*2i + (2^(1/2)/4 - 1/4)^(1/2)*2i) + atan((2^(1 
/2)*(- 2^(1/2)/4 - 1/4)^(1/2)*(x + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1 
/2)*(- 2^(1/2)/4 - 1/4)^(1/2) + 64) + (2^(1/2)*(2^(1/2)/4 - 1/4)^(1/2)*(x 
+ 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) + 6 
4))*((- 2^(1/2)/4 - 1/4)^(1/2)*2i - (2^(1/2)/4 - 1/4)^(1/2)*2i)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.39 \[ \int \frac {\sqrt {1+x} \left (1+x^3\right )}{1+x^2} \, dx=-\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}-2 \sqrt {x +1}}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )-\sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}-2 \sqrt {x +1}}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )+\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}+2 \sqrt {x +1}}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )+\sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}+2 \sqrt {x +1}}{\sqrt {\sqrt {2}-1}\, \sqrt {2}}\right )-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {x +1}\, \sqrt {\sqrt {2}+1}\, \sqrt {2}+\sqrt {2}+x +1\right )}{2}+\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {x +1}\, \sqrt {\sqrt {2}+1}\, \sqrt {2}+\sqrt {2}+x +1\right )}{2}+\frac {\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (-\sqrt {x +1}\, \sqrt {\sqrt {2}+1}\, \sqrt {2}+\sqrt {2}+x +1\right )}{2}-\frac {\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (\sqrt {x +1}\, \sqrt {\sqrt {2}+1}\, \sqrt {2}+\sqrt {2}+x +1\right )}{2}+\frac {2 \sqrt {x +1}\, x^{2}}{5}+\frac {2 \sqrt {x +1}\, x}{15}-\frac {34 \sqrt {x +1}}{15} \] Input:

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

Output:

( - 30*sqrt(sqrt(2) - 1)*sqrt(2)*atan((sqrt(sqrt(2) + 1)*sqrt(2) - 2*sqrt( 
x + 1))/(sqrt(sqrt(2) - 1)*sqrt(2))) - 30*sqrt(sqrt(2) - 1)*atan((sqrt(sqr 
t(2) + 1)*sqrt(2) - 2*sqrt(x + 1))/(sqrt(sqrt(2) - 1)*sqrt(2))) + 30*sqrt( 
sqrt(2) - 1)*sqrt(2)*atan((sqrt(sqrt(2) + 1)*sqrt(2) + 2*sqrt(x + 1))/(sqr 
t(sqrt(2) - 1)*sqrt(2))) + 30*sqrt(sqrt(2) - 1)*atan((sqrt(sqrt(2) + 1)*sq 
rt(2) + 2*sqrt(x + 1))/(sqrt(sqrt(2) - 1)*sqrt(2))) - 15*sqrt(sqrt(2) + 1) 
*sqrt(2)*log( - sqrt(x + 1)*sqrt(sqrt(2) + 1)*sqrt(2) + sqrt(2) + x + 1) + 
 15*sqrt(sqrt(2) + 1)*sqrt(2)*log(sqrt(x + 1)*sqrt(sqrt(2) + 1)*sqrt(2) + 
sqrt(2) + x + 1) + 15*sqrt(sqrt(2) + 1)*log( - sqrt(x + 1)*sqrt(sqrt(2) + 
1)*sqrt(2) + sqrt(2) + x + 1) - 15*sqrt(sqrt(2) + 1)*log(sqrt(x + 1)*sqrt( 
sqrt(2) + 1)*sqrt(2) + sqrt(2) + x + 1) + 12*sqrt(x + 1)*x**2 + 4*sqrt(x + 
 1)*x - 68*sqrt(x + 1))/30