Integrand size = 23, antiderivative size = 94 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \text {arctanh}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {929, 272, 52, 65, 213} \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=-\frac {2 \sqrt {x+1} \sqrt {x^2-x+1} \text {arctanh}\left (\sqrt {x^3+1}\right )}{3 \sqrt {x^3+1}}+\frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1}+\frac {2}{9} \sqrt {x+1} \sqrt {x^2-x+1} \left (x^3+1\right ) \]
[In]
[Out]
Rule 52
Rule 65
Rule 213
Rule 272
Rule 929
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {\left (1+x^3\right )^{3/2}}{x} \, dx}{\sqrt {1+x^3}} \\ & = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2}}{x} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )+\frac {\left (2 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \\ & = \frac {2}{3} \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2}{9} \sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )-\frac {2 \sqrt {1+x} \sqrt {1-x+x^2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )}{3 \sqrt {1+x^3}} \\ \end{align*}
Time = 10.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\frac {2}{9} \left (\sqrt {1+x} \sqrt {1-x+x^2} \left (4+x^3\right )-3 \text {arctanh}\left (\sqrt {1+x} \sqrt {1-x+x^2}\right )\right ) \]
[In]
[Out]
Time = 0.78 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {2 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (-x^{3} \sqrt {x^{3}+1}+3 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )-4 \sqrt {x^{3}+1}\right )}{9 \sqrt {x^{3}+1}}\) | \(57\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x^{3} \sqrt {x^{3}+1}}{9}+\frac {8 \sqrt {x^{3}+1}}{9}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\right )}{x^{3}+1}\) | \(70\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\frac {2}{9} \, {\left (x^{3} + 4\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} - \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x^{2} - x + 1} \sqrt {x + 1} - 1\right ) \]
[In]
[Out]
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x} \, dx=\int \frac {{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}}{x} \,d x \]
[In]
[Out]