Optimal. Leaf size=313 \[ -\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{9 \sqrt{a} b^{2/3}} \]
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Rubi [A] time = 0.320473, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2139, 219, 2138, 203} \[ -\frac{2 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\sqrt [3]{a} f+\sqrt [3]{b} e\right ) F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} b^{2/3} \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{b x^3-a}}-\frac{2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{b x^3-a}}\right )}{9 \sqrt{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2139
Rule 219
Rule 2138
Rule 203
Rubi steps
\begin{align*} \int \frac{e+f x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{-a+b x^3}} \, dx &=\frac{1}{6} \left (\frac{e}{\sqrt [3]{a}}-\frac{2 f}{\sqrt [3]{b}}\right ) \int \frac{2 \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{-a+b x^3}} \, dx+\frac{1}{3} \left (\frac{e}{\sqrt [3]{a}}+\frac{f}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt{-a+b x^3}} \, dx\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (\frac{e}{\sqrt [3]{a}}+\frac{f}{\sqrt [3]{b}}\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 \sqrt [4]{3} \sqrt [3]{b} \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 (2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{9+a x^2} \, dx,x,\frac{\left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt{-a+b x^3}}\right )}{3 b^{2/3}}\\ &=-\frac{2 \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \tan ^{-1}\left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt{-a+b x^3}}\right )}{9 \sqrt{a} b^{2/3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (\frac{e}{\sqrt [3]{a}}+\frac{f}{\sqrt [3]{b}}\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 \sqrt [4]{3} \sqrt [3]{b} \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 = 0.88537, size = 448, normalized size = 1.43 \[ \frac{2 \sqrt{\frac{\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (-i \sqrt{-\frac{i \left (2 \sqrt [3]{a}+\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x\right )}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \sqrt{\frac{b^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \left (\sqrt [3]{b} e-2 \sqrt [3]{a} f\right ) \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )-\frac{1}{2} i f \sqrt{\frac{\left (\sqrt{3}-i\right ) \sqrt [3]{a}+\left (\sqrt{3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}}} \left (\left (\sqrt{3}-3 i\right ) \sqrt [3]{a}-\left (\sqrt{3}+3 i\right ) \sqrt [3]{b} x\right ) F\left (\sin ^{-1}\left (\sqrt{-\frac{i \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt{3}\right ) \sqrt [3]{a}}}\right )|\frac{1}{2} \left (1+i \sqrt{3}\right )\right )\right )}{\left (\sqrt [3]{-1}-2\right ) b^{2/3} \sqrt{\frac{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3-a}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{(fx+e) \left ( 2\,\sqrt [3]{a}+\sqrt [3]{b}x \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 + 2 \, a^{\frac{1}{3}}\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}{\left (2 \sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{- a + b x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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