Optimal. Leaf size=503 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} d+\sqrt [3]{b} c\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} d \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{2 d \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]
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Rubi [A] time = 0.148506, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1878, 218, 1877} \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a} d+\sqrt [3]{b} c\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} d \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt{a-b x^3}}+\frac{2 d \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )} \]
Antiderivative was successfully verified.
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Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{c+d x}{\sqrt{a-b x^3}} \, dx &=-\frac{d \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt{a-b x^3}} \, dx}{\sqrt [3]{b}}-\left (-c-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt{a-b x^3}} \, dx\\ &=\frac{2 d \sqrt{a-b x^3}}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} d \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}} E\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 )}{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{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{b} c+\left (1-\sqrt{3}\right ) \sqrt [3]{a} d\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 )}{\sqrt [4]{3} 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 = 0.0354163, size = 75, normalized size = 0.15 \[ \frac{x \sqrt{1-\frac{b x^3}{a}} \left (2 c \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\frac{b x^3}{a}\right )+d x \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b x^3}{a}\right )\right )}{2 \sqrt{a-b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 681, normalized size = 1.4 \begin{align*} \text{result too large to display} \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{d x + c}{\sqrt{-b x^{3} + a}}\,{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{\sqrt{-b x^{3} + a}{\left (d x + c\right )}}{b x^{3} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.72232, size = 82, normalized size = 0.16 \begin{align*} \frac{c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{4}{3}\right )} + \frac{d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{5}{3}\right )} \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{d x + c}{\sqrt{-b x^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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