Optimal. Leaf size=278 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (55 a e^2-22 b d e+16 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \sqrt{d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e} \]
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Rubi [A] time = 0.182336, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1411, 388, 218} \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (55 a e^2-22 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \sqrt{d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e} \]
Antiderivative was successfully verified.
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Rule 1411
Rule 388
Rule 218
Rubi steps
\begin{align*} \int \frac{a+b x^3+c x^6}{\sqrt{d+e x^3}} \, dx &=\frac{2 c x^4 \sqrt{d+e x^3}}{11 e}+\frac{2 \int \frac{\frac{11 a e}{2}-\left (4 c d-\frac{11 b e}{2}\right ) x^3}{\sqrt{d+e x^3}} \, dx}{11 e}\\ &=-\frac{2 (8 c d-11 b e) x \sqrt{d+e x^3}}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e}-\frac{1}{55} \left (-55 a-\frac{2 d (8 c d-11 b e)}{e^2}\right ) \int \frac{1}{\sqrt{d+e x^3}} \, dx\\ &=-\frac{2 (8 c d-11 b e) x \sqrt{d+e x^3}}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e}+\frac{2 \sqrt{2+\sqrt{3}} \left (16 c d^2-22 b d e+55 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}\\ \end{align*}
Mathematica [C] time = 0.0905538, size = 98, normalized size = 0.35 \[ \frac{x \left (\sqrt{\frac{e x^3}{d}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right ) \left (11 e (5 a e-2 b d)+16 c d^2\right )-2 \left (d+e x^3\right ) \left (-11 b e+8 c d-5 c e x^3\right )\right )}{55 e^2 \sqrt{d+e x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 907, normalized size = 3.3 \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{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}\,{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{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.62862, size = 119, normalized size = 0.43 \begin{align*} \frac{a 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{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{4}{3}\right )} + \frac{b x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{7}{3}\right )} + \frac{c x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{10}{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{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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