Optimal. Leaf size=476 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c+\sqrt{3} d+d\right )} \]
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Rubi [A] time = 1.10559, antiderivative size = 476, 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, Rules used = {2144, 218, 2142, 2113, 537, 571, 93, 208} \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3} \left (c+\sqrt{3} d+d\right )} \]
Antiderivative was successfully verified.
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Rule 2144
Rule 218
Rule 2142
Rule 2113
Rule 537
Rule 571
Rule 93
Rule 208
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*}
Mathematica [C] time = 0.6949, size = 233, normalized size = 0.49 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{3 f \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} (c f-d e) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1} d-c}\right )}{3 d \sqrt{1-x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.007, size = 265, normalized size = 0.6 \begin{align*}{\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{x-1}{-{\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{x-1}{-{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-x^{3} + 1}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e + f x}{\sqrt{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (c + d x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-x^{3} + 1}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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