Integrand size = 27, antiderivative size = 321 \[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=-\frac {\left (c-\left (1+\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \arctan \left (\frac {\sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\left (c-\left (1-\sqrt {3}\right ) d\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
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Time = 0.73 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2167, 2138, 551, 585, 95, 211} \[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \left (c-\left (1-\sqrt {3}\right ) d\right )}-\frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (c-\left (1+\sqrt {3}\right ) d\right ) \arctan \left (\frac {\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {c^2+c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {c-d}}\right )}{\sqrt {d} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \sqrt {c-d} \sqrt {c^2+c d+d^2}} \]
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Rule 95
Rule 211
Rule 551
Rule 585
Rule 2138
Rule 2167
Rubi steps \begin{align*} \text {integral}& = \frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-c+\left (1-\sqrt {3}\right ) d+\left (-c+\left (1+\sqrt {3}\right ) d\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \\ & = -\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (-c+d+\sqrt {3} d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-c+\left (1-\sqrt {3}\right ) d\right )^2-\left (-c+\left (1+\sqrt {3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (-c+\left (1-\sqrt {3}\right ) d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-c+\left (1-\sqrt {3}\right ) d\right )^2-\left (-c+\left (1+\sqrt {3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \\ & = \frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\left (c-\left (1-\sqrt {3}\right ) d\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {\left (2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (-c+d+\sqrt {3} d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (-c+\left (1-\sqrt {3}\right ) d\right )^2-\left (-c+\left (1+\sqrt {3}\right ) d\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \\ & = \frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\left (c-\left (1-\sqrt {3}\right ) d\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (-c+d+\sqrt {3} d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{-\left (-c+\left (1-\sqrt {3}\right ) d\right )^2+\left (-c+\left (1+\sqrt {3}\right ) d\right )^2-\left (\left (-c+\left (1-\sqrt {3}\right ) d\right )^2+\left (7-4 \sqrt {3}\right ) \left (-c+\left (1+\sqrt {3}\right ) d\right )^2\right ) x^2} \, dx,x,\frac {\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {-\frac {\left (-2+\sqrt {3}\right ) \left (1-x+x^2\right )}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \\ & = -\frac {\left (c-d-\sqrt {3} d\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tan ^{-1}\left (\frac {\sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\left (c-\left (1-\sqrt {3}\right ) d\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.54 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.73 \[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c-\left (3+\sqrt {3}\right ) d\right ) \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{3 d \sqrt {-1-x^3}} \]
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Time = 1.29 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}-\frac {2 i \left (d \sqrt {3}-c +d \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(266\) |
elliptic | \(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}-\frac {2 i \left (d \sqrt {3}-c +d \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\) | \(266\) |
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Exception generated. \[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\int \frac {x + 1 + \sqrt {3}}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \]
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\[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {-x^{3} - 1} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {-x^{3} - 1} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx=\text {Hanged} \]
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