Optimal. Leaf size=475 \[ -\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac {\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c^2-c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c+d}}\right )}{\sqrt {d} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \sqrt {c+d} \sqrt {c^2-c d+d^2}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \Pi \left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2};\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \left (c^2+2 c d-2 d^2\right )}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \left (c+\sqrt {3} d+d\right )} \]
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Rubi [A] time = 0.92, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2144, 219, 2142, 2113, 537, 571, 93, 208} \[ -\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac {\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c^2-c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c+d}}\right )}{\sqrt {d} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \sqrt {c+d} \sqrt {c^2-c d+d^2}}+\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} (d e-c f) \Pi \left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2};-\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \left (c^2+2 c d-2 d^2\right )}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (e+\sqrt {3} f+f\right ) F\left (\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \left (c+\sqrt {3} d+d\right )} \]
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rule 219
Rule 537
Rule 571
Rule 2113
Rule 2142
Rule 2144
Rubi steps
\begin {align*} \int \frac {e+f x}{(c+d x) \sqrt {-1+x^3}} \, dx &=\frac {\left (e+f+\sqrt {3} f\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx}{c+d+\sqrt {3} d}+\frac {(d e-c f) \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx}{c+d+\sqrt {3} d}\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (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 )}{\sqrt [4]{3} \left (c+d+\sqrt {3} 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}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \operatorname {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 )}{\left (c+d+\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (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 )}{\sqrt [4]{3} \left (c+d+\sqrt {3} 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}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \operatorname {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+d-\sqrt {3} d\right ) (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \operatorname {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 )}{\left (c+d+\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ &=-\frac {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (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 )}{\sqrt [4]{3} \left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \left (c^2+2 c d-2 d^2\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}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \operatorname {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 {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (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 )}{\sqrt [4]{3} \left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \left (c^2+2 c d-2 d^2\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}} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \operatorname {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 {(d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (e+f+\sqrt {3} f\right ) (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 )}{\sqrt [4]{3} \left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \left (c^2+2 c d-2 d^2\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 231, normalized size = 0.49 \[ \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {3 f \left (x+\sqrt [3]{-1}\right ) \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \sqrt {3} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} (c f-d e) \Pi \left (\frac {i \sqrt {3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1} d-c}\right )}{3 d \sqrt {x^3-1}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f x + e}{\sqrt {x^{3} - 1} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 274, normalized size = 0.58 \[ \frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, f \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, d}+\frac {2 \left (-c f +d e \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {c}{d}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (\frac {c}{d}+1\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f x + e}{\sqrt {x^{3} - 1} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 355, normalized size = 0.75 \[ -\frac {2\,f\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (c\,f-d\,e\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {c}{d}+1};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d^2\,\left (\frac {c}{d}+1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e + f x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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