Optimal. Leaf size=681 \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-1870 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{85085 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{24 a^2 e \sqrt{a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{91 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 a \sqrt{a+b x^3} (5 b c-2 a f)}{45 b^2}+\frac{6 a x \sqrt{a+b x^3} (17 b d-8 a g)}{935 b^2}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b} \]
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Rubi [A] time = 1.42152, antiderivative size = 681, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {1826, 1836, 1888, 1594, 1886, 261, 1878, 218, 1877} \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-1870 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{85085 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{24 a^2 e \sqrt{a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{91 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 a \sqrt{a+b x^3} (5 b c-2 a f)}{45 b^2}+\frac{6 a x \sqrt{a+b x^3} (17 b d-8 a g)}{935 b^2}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b} \]
Antiderivative was successfully verified.
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Rule 1826
Rule 1836
Rule 1888
Rule 1594
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{1}{2} (3 a) \int \frac{x^2 \left (\frac{2 c}{9}+\frac{2 d x}{11}+\frac{2 e x^2}{13}+\frac{2 f x^3}{15}+\frac{2 g x^4}{17}\right )}{\sqrt{a+b x^3}} \, dx\\ &=\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(3 a) \int \frac{x^2 \left (\frac{11 b c}{9}+\frac{1}{17} (17 b d-8 a g) x+\frac{11}{13} b e x^2+\frac{11}{15} b f x^3\right )}{\sqrt{a+b x^3}} \, dx}{11 b}\\ &=\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(2 a) \int \frac{x^2 \left (\frac{11}{10} b (5 b c-2 a f)+\frac{9}{34} b (17 b d-8 a g) x+\frac{99}{26} b^2 e x^2\right )}{\sqrt{a+b x^3}} \, dx}{33 b^2}\\ &=\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(4 a) \int \frac{-\frac{99}{13} a b^2 e x+\frac{77}{20} b^2 (5 b c-2 a f) x^2+\frac{63}{68} b^2 (17 b d-8 a g) x^3}{\sqrt{a+b x^3}} \, dx}{231 b^3}\\ &=\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(4 a) \int \frac{x \left (-\frac{99}{13} a b^2 e+\frac{77}{20} b^2 (5 b c-2 a f) x+\frac{63}{68} b^2 (17 b d-8 a g) x^2\right )}{\sqrt{a+b x^3}} \, dx}{231 b^3}\\ &=\frac{6 a (17 b d-8 a g) x \sqrt{a+b x^3}}{935 b^2}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(8 a) \int \frac{-\frac{63}{68} a b^2 (17 b d-8 a g)-\frac{495}{26} a b^3 e x+\frac{77}{8} b^3 (5 b c-2 a f) x^2}{\sqrt{a+b x^3}} \, dx}{1155 b^4}\\ &=\frac{6 a (17 b d-8 a g) x \sqrt{a+b x^3}}{935 b^2}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{(8 a) \int \frac{-\frac{63}{68} a b^2 (17 b d-8 a g)-\frac{495}{26} a b^3 e x}{\sqrt{a+b x^3}} \, dx}{1155 b^4}+\frac{(a (5 b c-2 a f)) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{15 b}\\ &=\frac{2 a (5 b c-2 a f) \sqrt{a+b x^3}}{45 b^2}+\frac{6 a (17 b d-8 a g) x \sqrt{a+b x^3}}{935 b^2}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}-\frac{\left (12 a^2 e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{91 b^{4/3}}-\frac{\left (6 a^2 \left (1547 b d-1870 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{85085 b^2}\\ &=\frac{2 a (5 b c-2 a f) \sqrt{a+b x^3}}{45 b^2}+\frac{6 a (17 b d-8 a g) x \sqrt{a+b x^3}}{935 b^2}+\frac{6 a e x^2 \sqrt{a+b x^3}}{91 b}+\frac{2 a f x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a g x^4 \sqrt{a+b x^3}}{187 b}-\frac{24 a^2 e \sqrt{a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 x^2 \sqrt{a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} e \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 )}{91 b^{5/3} \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{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (1547 b d-1870 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g\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 )}{85085 b^{7/3} \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 = 0.253586, size = 158, normalized size = 0.23 \[ \frac{2 \sqrt{a+b x^3} \left (-\left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} \left (26 a (187 f+180 g x)-b \left (12155 c+9945 d x+33 x^2 (255 e+13 x (17 f+15 g x))\right )\right )+585 a x (8 a g-17 b d) \, _2F_1\left (-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-8415 a b e x^2 \, _2F_1\left (-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{109395 b^2 \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 1197, normalized size = 1.8 \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*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} c}{9 \, b} + \int{\left (g x^{6} + f x^{5} + e x^{4} + d x^{3}\right )} \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 ({\left (g x^{6} + f x^{5} + e x^{4} + d x^{3} + c x^{2}\right )} \sqrt{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 = 4.52438, size = 223, normalized size = 0.33 \begin{align*} \frac{\sqrt{a} d 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{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{\sqrt{a} e x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{\sqrt{a} g 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{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + c \left (\begin{cases} \frac{\sqrt{a} x^{3}}{3} & \text{for}\: b = 0 \\\frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b} & \text{otherwise} \end{cases}\right ) + f \left (\begin{cases} - \frac{4 a^{2} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 a x^{3} \sqrt{a + b x^{3}}}{45 b} + \frac{2 x^{6} \sqrt{a + b x^{3}}}{15} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\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 (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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