Integrand size = 23, antiderivative size = 175 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\frac {9}{10} x \sqrt {1+x} \sqrt {1-x+x^2}-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{10 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \]
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Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {929, 283, 201, 224} \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{10 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {9}{10} x \sqrt {x^2-x+1} \sqrt {x+1}-\frac {\sqrt {x^2-x+1} \left (x^3+1\right ) \sqrt {x+1}}{2 x^2} \]
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Rule 201
Rule 224
Rule 283
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^3} \, dx}{\sqrt {1+x^3}} \\ & = -\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {\left (9 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \sqrt {1+x^3} \, dx}{4 \sqrt {1+x^3}} \\ & = \frac {9}{10} x \sqrt {1+x} \sqrt {1-x+x^2}-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {\left (27 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{20 \sqrt {1+x^3}} \\ & = \frac {9}{10} x \sqrt {1+x} \sqrt {1-x+x^2}-\frac {\sqrt {1+x} \sqrt {1-x+x^2} \left (1+x^3\right )}{2 x^2}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \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 )}{10 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {1+x} \left (\frac {2 \left (1-x+x^2\right ) \left (-5+4 x^3\right )}{x^2}-\frac {27 i \sqrt {2} \sqrt {\frac {i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}} \sqrt {\frac {-i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i (1+x)}{3 i+\sqrt {3}}}}\right )}{20 \sqrt {1-x+x^2}} \]
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Time = 0.67 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (4 x^{3}-5\right )}{10 x^{2}}+\frac {27 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{10 \sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(173\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {27 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{10 \sqrt {x^{3}+1}}-\frac {\sqrt {x^{3}+1}}{2 x^{2}}\right )}{x^{3}+1}\) | \(176\) |
default | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (27 i \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) \sqrt {3}\, x^{2}-81 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x^{2}-8 x^{6}+2 x^{3}+10\right )}{20 \left (x^{3}+1\right ) x^{2}}\) | \(264\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.22 \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\frac {27 \, x^{2} {\rm weierstrassPInverse}\left (0, -4, x\right ) + {\left (4 \, x^{3} - 5\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{10 \, x^{2}} \]
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\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}} \left (x^{2} - x + 1\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
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\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
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\[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (x^{2} - x + 1\right )}^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (x+1\right )}^{3/2}\,{\left (x^2-x+1\right )}^{3/2}}{x^3} \,d x \]
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