Optimal. Leaf size=396 \[ \frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (667 a e^2-58 b d e+16 c d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt{3}\right )}{124729 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}+\frac{2 x \left (d+e x^3\right )^{5/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{11339 e^2}+\frac{30 d x \left (d+e x^3\right )^{3/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54 d^2 x \sqrt{d+e x^3} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}-\frac{2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e} \]
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Rubi [A] time = 0.41577, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1411, 388, 195, 218} \[ \frac{2 x \left (d+e x^3\right )^{5/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{11339 e^2}+\frac{30 d x \left (d+e x^3\right )^{3/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54 d^2 x \sqrt{d+e x^3} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (667 a e^2-58 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{124729 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e} \]
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 )^{5/2} \left (a+b x^3+c x^6\right ) \, dx &=\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac{2 \int \left (d+e x^3\right )^{5/2} \left (\frac{29 a e}{2}-\left (4 c d-\frac{29 b e}{2}\right ) x^3\right ) \, dx}{29 e}\\ &=-\frac{2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}-\frac{1}{667} \left (-667 a-\frac{2 d (8 c d-29 b e)}{e^2}\right ) \int \left (d+e x^3\right )^{5/2} \, dx\\ &=\frac{2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac{2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac{\left (15 d \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \left (d+e x^3\right )^{3/2} \, dx}{11339}\\ &=\frac{30 d \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac{2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac{2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac{\left (135 d^2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \sqrt{d+e x^3} \, dx}{124729}\\ &=\frac{54 d^2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \sqrt{d+e x^3}}{124729}+\frac{30 d \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac{2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac{2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac{\left (81 d^3 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right )\right ) \int \frac{1}{\sqrt{d+e x^3}} \, dx}{124729}\\ &=\frac{54 d^2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \sqrt{d+e x^3}}{124729}+\frac{30 d \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{3/2}}{124729}+\frac{2 \left (667 a+\frac{2 d (8 c d-29 b e)}{e^2}\right ) x \left (d+e x^3\right )^{5/2}}{11339}-\frac{2 (8 c d-29 b e) x \left (d+e x^3\right )^{7/2}}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left (16 c d^2-58 b d e+667 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 )}{124729 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 = 0.178911, size = 103, normalized size = 0.26 \[ \frac{x \sqrt{d+e x^3} \left (\frac{\, _2F_1\left (-\frac{5}{2},\frac{1}{3};\frac{4}{3};-\frac{e x^3}{d}\right ) \left (29 d^2 e (23 a e-2 b d)+16 c d^4\right )}{\sqrt{\frac{e x^3}{d}+1}}-2 \left (d+e x^3\right )^3 \left (-29 b e+8 c d-23 c e x^3\right )\right )}{667 e^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.161, size = 1070, normalized size = 2.7 \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{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 ({\left (c e^{2} x^{12} +{\left (2 \, c d e + b e^{2}\right )} x^{9} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{6} +{\left (b d^{2} + 2 \, a d e\right )} x^{3} + a d^{2}\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 = 10.0827, size = 400, normalized size = 1.01 \begin{align*} \text{result too large to display} \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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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