Optimal. Leaf size=356 \[ \frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt{3}\right )}{21505 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}+\frac{2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac{18 d x \sqrt{d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}-\frac{2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]
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Rubi [A] time = 0.311484, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1411, 388, 195, 218} \[ \frac{2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac{18 d x \sqrt{d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{21505 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]
Antiderivative was successfully verified.
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Rule 1411
Rule 388
Rule 195
Rule 218
Rubi steps
\begin{align*} \int \left (d+e x^3\right )^{3/2} \left (a+b x^3+c x^6\right ) \, dx &=\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac{2 \int \left (d+e x^3\right )^{3/2} \left (\frac{23 a e}{2}-\left (4 c d-\frac{23 b e}{2}\right ) x^3\right ) \, dx}{23 e}\\ &=-\frac{2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}-\frac{1}{391} \left (-391 a-\frac{2 d (8 c d-23 b e)}{e^2}\right ) \int \left (d+e x^3\right )^{3/2} \, dx\\ &=\frac{2 \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac{2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac{\left (9 d \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \sqrt{d+e x^3} \, dx}{4301}\\ &=\frac{18 d \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt{d+e x^3}}{21505}+\frac{2 \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac{2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac{\left (27 d^2 \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right )\right ) \int \frac{1}{\sqrt{d+e x^3}} \, dx}{21505}\\ &=\frac{18 d \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right ) x \sqrt{d+e x^3}}{21505}+\frac{2 \left (391 a+\frac{2 d (8 c d-23 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{4301}-\frac{2 (8 c d-23 b e) x \left (d+e x^3\right )^{5/2}}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left (16 c d^2-46 b d e+391 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 )}{21505 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 = 0.153007, size = 101, normalized size = 0.28 \[ \frac{x \sqrt{d+e x^3} \left (\frac{\, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{e x^3}{d}\right ) \left (23 d e (17 a e-2 b d)+16 c d^3\right )}{\sqrt{\frac{e x^3}{d}+1}}-2 \left (d+e x^3\right )^2 \left (-23 b e+8 c d-17 c e x^3\right )\right )}{391 e^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 1010, normalized size = 2.8 \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{\left (c x^{6} + b x^{3} + a\right )}{\left (e x^{3} + d\right )}^{\frac{3}{2}}\,{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 ({\left (c e x^{9} +{\left (c d + b e\right )} x^{6} +{\left (b d + a e\right )} x^{3} + a d\right )} \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 = 5.8478, size = 257, normalized size = 0.72 \begin{align*} \frac{a d^{\frac{3}{2}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{a \sqrt{d} e 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 \Gamma \left (\frac{7}{3}\right )} + \frac{b d^{\frac{3}{2}} 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 \Gamma \left (\frac{7}{3}\right )} + \frac{b \sqrt{d} e 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 \Gamma \left (\frac{10}{3}\right )} + \frac{c d^{\frac{3}{2}} 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 \Gamma \left (\frac{10}{3}\right )} + \frac{c \sqrt{d} e x^{10} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{13}{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{\left (c x^{6} + b x^{3} + a\right )}{\left (e x^{3} + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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