Optimal. Leaf size=336 \[ \frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
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Rubi [A] time = 0.567915, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2141, 218, 2140, 206} \[ \frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} f+\sqrt [3]{b} e\right ) \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2141
Rule 218
Rule 2140
Rule 206
Rubi steps
\begin{align*} \int \frac{e+f x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx &=-\frac{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (22 a b-\left (1-\sqrt{3}\right )^3 a b\right )+6 a b^{4/3} x}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{a-b x^3}} \, dx}{12 \sqrt{3} a^{4/3} b^{4/3}}+\frac{\left (6 a b^{4/3} e-\left (1-\sqrt{3}\right ) \sqrt [3]{a} \left (22 a b-\left (1-\sqrt{3}\right )^3 a b\right ) f\right ) \int \frac{1}{\sqrt{a-b x^3}} \, dx}{\left (1-\sqrt{3}\right ) \sqrt [3]{a} \sqrt [3]{b} \left (28 a b-\left (1-\sqrt{3}\right )^3 a b\right )}\\ &=\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{b} e+\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \operatorname{Subst}\left (\int \frac{1}{1+\left (3-2 \sqrt{3}\right ) a x^2} \, dx,x,\frac{1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{a-b x^3}}\right )}{\sqrt{3} b^{2/3}}\\ &=\frac{\left (\sqrt [3]{b} e+\left (1-\sqrt{3}\right ) \sqrt [3]{a} f\right ) \tanh ^{-1}\left (\frac{\sqrt{-3+2 \sqrt{3}} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt{a-b x^3}}\right )}{\sqrt{3 \left (-3+2 \sqrt{3}\right )} \sqrt{a} b^{2/3}}+\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{b} e+\left (1+\sqrt{3}\right ) \sqrt [3]{a} f\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt [3]{a} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}\\ \end{align*}
Mathematica [C] time = 1.64916, size = 466, normalized size = 1.39 \[ -\frac{4 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac{1}{2} f \left (i \left (-3+(2+i) \sqrt{3}\right ) \sqrt [3]{a}+\left (3-(2-i) \sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{\frac{\left (\sqrt{3}-i\right ) \sqrt [3]{a}+\left (\sqrt{3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-i \sqrt{-\frac{i \left (2 \sqrt [3]{a}+\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x\right )}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\sqrt [3]{b} e-\left (\sqrt{3}-1\right ) \sqrt [3]{a} f\right ) \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{\left (3-(2-i) \sqrt{3}\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{a-b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{(fx+e) \left ( -\sqrt [3]{b}x+\sqrt [3]{a} \left ( 1-\sqrt{3} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-b{x}^{3}+a}}}}\, dx \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{-b x^{3} + a}{\left (b^{\frac{1}{3}} x + a^{\frac{1}{3}}{\left (\sqrt{3} - 1\right )}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{2 \,{\left (2 \, b f x^{4} + 2 \, b e x^{3} - 2 \, a f x - 2 \, a e - \sqrt{3}{\left (b f x^{4} + b e x^{3} + 2 \, a f x + 2 \, a e\right )}\right )} \sqrt{-b x^{3} + a} a^{\frac{2}{3}} +{\left (b f x^{5} + b e x^{4} + 8 \, a f x^{2} + 8 \, a e x - \sqrt{3}{\left (b f x^{5} + b e x^{4} - 4 \, a f x^{2} - 4 \, a e x\right )}\right )} \sqrt{-b x^{3} + a} a^{\frac{1}{3}} b^{\frac{1}{3}} +{\left (b f x^{6} + b e x^{5} - 10 \, a f x^{3} - 10 \, a e x^{2} - 6 \, \sqrt{3}{\left (a f x^{3} + a e x^{2}\right )}\right )} \sqrt{-b x^{3} + a} b^{\frac{2}{3}}}{b^{3} x^{9} - 21 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 8 \, a^{3}}, x\right ) \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}{- \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt{3} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, dx - \int \frac{f x}{- \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt{3} \sqrt [3]{a} \sqrt{a - b x^{3}} + \sqrt [3]{b} x \sqrt{a - b x^{3}}}\, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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