Optimal. Leaf size=309 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (e (7 a e+2 b d)+16 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} d^2 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}} \]
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Rubi [A] time = 0.210659, antiderivative size = 309, 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 = {1409, 385, 218} \[ -\frac{2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (e (7 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} d^2 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]
Antiderivative was successfully verified.
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Rule 1409
Rule 385
Rule 218
Rubi steps
\begin{align*} \int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (2 c d^2-e (2 b d+7 a e)\right )-\frac{9}{2} c d e x^3}{\left (d+e x^3\right )^{3/2}} \, dx}{9 d e^2}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac{2 \left (11 c d^2-2 b d e-7 a e^2\right ) x}{27 d^2 e^2 \sqrt{d+e x^3}}-\frac{\left (4 \left (-\frac{9}{2} c d^2 e+\frac{1}{4} e \left (2 c d^2-e (2 b d+7 a e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x^3}} \, dx}{27 d^2 e^3}\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) x}{9 d e^2 \left (d+e x^3\right )^{3/2}}-\frac{2 \left (11 c d^2-2 b d e-7 a e^2\right ) x}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left (16 c d^2+e (2 b d+7 a e)\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 )}{27 \sqrt [4]{3} d^2 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 = 0.139048, size = 129, normalized size = 0.42 \[ \frac{x \left (d+e x^3\right ) \sqrt{\frac{e x^3}{d}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{e x^3}{d}\right ) \left (e (7 a e+2 b d)+16 c d^2\right )-2 x \left (e \left (b d \left (d-2 e x^3\right )-a e \left (10 d+7 e x^3\right )\right )+c d^2 \left (8 d+11 e x^3\right )\right )}{27 d^2 e^2 \left (d+e x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 1005, 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}{{\left (e x^{3} + d\right )}^{\frac{5}{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 (\frac{{\left (c x^{6} + b x^{3} + a\right )} \sqrt{e x^{3} + d}}{e^{3} x^{9} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{3} + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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