Optimal. Leaf size=465 \[ \frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (d e-c f) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c^2-2 c d-2 d^2\right )}+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\sqrt{3} f-f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )} \]
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Rubi [A] time = 3.02387, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (d e-c f) \tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c-\left (1+\sqrt{3}\right ) d\right )^2}{\left (c-\left (1-\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c^2-2 c d-2 d^2\right )}+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\sqrt{3} f-f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )} \]
Warning: Unable to verify antiderivative.
[In] Int[(e + f*x)/((c + d*x)*Sqrt[-1 - x^3]),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/(d*x+c)/(-x**3-1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.935545, size = 213, normalized size = 0.46 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{f \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} (c f-d e) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e + f*x)/((c + d*x)*Sqrt[-1 - x^3]),x]
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Maple [A] time = 0.011, size = 265, normalized size = 0.6 \[{\frac{-{\frac{2\,i}{3}}f\sqrt{3}}{d}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{{\frac{2\,i}{3}} \left ( -cf+de \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/(d*x+c)/(-x^3-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(d*x + c)),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)/(d*x+c)/(-x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(d*x + c)),x, algorithm="giac")
[Out]