Integrand size = 23, antiderivative size = 146 \[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {-1-x^3}{2 x^2 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {\sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {929, 331, 224} \[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {\sqrt {2+\sqrt {3}} \sqrt {x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^2-x+1}}-\frac {x^3+1}{2 x^2 \sqrt {x+1} \sqrt {x^2-x+1}} \]
[In]
[Out]
Rule 224
Rule 331
Rule 929
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^3} \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx}{\sqrt {1+x} \sqrt {1-x+x^2}} \\ & = -\frac {1+x^3}{2 x^2 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {\sqrt {1+x^3} \int \frac {1}{\sqrt {1+x^3}} \, dx}{4 \sqrt {1+x} \sqrt {1-x+x^2}} \\ & = -\frac {1+x^3}{2 x^2 \sqrt {1+x} \sqrt {1-x+x^2}}-\frac {\sqrt {2+\sqrt {3}} \sqrt {1+x} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1-x+x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.41 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\frac {-\frac {6 \sqrt {1+x} \left (1-x+x^2\right )}{x^2}-\frac {i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {6-\frac {36 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}}}}{12 \sqrt {1-x+x^2}} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.09
method | result | size |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (-\frac {\sqrt {x^{3}+1}}{2 x^{2}}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(159\) |
risch | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}}{2 x^{2}}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{2 \sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(166\) |
default | \(\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (i \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x^{2}-3 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{2}-2 x^{3}-2\right )}{4 \left (x^{3}+1\right ) x^{2}}\) | \(259\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.21 \[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=-\frac {x^{2} {\rm weierstrassPInverse}\left (0, -4, x\right ) + \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{2 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^3 \sqrt {1+x} \sqrt {1-x+x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {x+1}\,\sqrt {x^2-x+1}} \,d x \]
[In]
[Out]