Integrand size = 35, antiderivative size = 733 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}-\frac {24 a^2 (19 b d-10 a g) \sqrt {a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-10 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{1616615 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 1.32 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1840, 1850, 1902, 1608, 1900, 267, 1892, 224, 1891} \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-10 a g) E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {24 a^2 \sqrt {a+b x^3} (19 b d-10 a g)}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right )}{1616615 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {6 a x \sqrt {a+b x^3} (17 b c-8 a f)}{935 b^2}+\frac {6 a x^2 \sqrt {a+b x^3} (19 b d-10 a g)}{1729 b^2}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b} \]
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Rule 224
Rule 267
Rule 1608
Rule 1840
Rule 1850
Rule 1891
Rule 1892
Rule 1900
Rule 1902
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {1}{2} (3 a) \int \frac {x^3 \left (\frac {2 c}{11}+\frac {2 d x}{13}+\frac {2 e x^2}{15}+\frac {2 f x^3}{17}+\frac {2 g x^4}{19}\right )}{\sqrt {a+b x^3}} \, dx \\ & = \frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(3 a) \int \frac {x^3 \left (\frac {13 b c}{11}+\frac {1}{19} (19 b d-10 a g) x+\frac {13}{15} b e x^2+\frac {13}{17} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{13 b} \\ & = \frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(6 a) \int \frac {x^3 \left (\frac {13}{34} b (17 b c-8 a f)+\frac {11}{38} b (19 b d-10 a g) x+\frac {143}{30} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{143 b^2} \\ & = \frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(4 a) \int \frac {-\frac {143}{10} a b^2 e x^2+\frac {117}{68} b^2 (17 b c-8 a f) x^3+\frac {99}{76} b^2 (19 b d-10 a g) x^4}{\sqrt {a+b x^3}} \, dx}{429 b^3} \\ & = \frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(4 a) \int \frac {x^2 \left (-\frac {143}{10} a b^2 e+\frac {117}{68} b^2 (17 b c-8 a f) x+\frac {99}{76} b^2 (19 b d-10 a g) x^2\right )}{\sqrt {a+b x^3}} \, dx}{429 b^3} \\ & = \frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(8 a) \int \frac {-\frac {99}{38} a b^2 (19 b d-10 a g) x-\frac {1001}{20} a b^3 e x^2+\frac {819}{136} b^3 (17 b c-8 a f) x^3}{\sqrt {a+b x^3}} \, dx}{3003 b^4} \\ & = \frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(8 a) \int \frac {x \left (-\frac {99}{38} a b^2 (19 b d-10 a g)-\frac {1001}{20} a b^3 e x+\frac {819}{136} b^3 (17 b c-8 a f) x^2\right )}{\sqrt {a+b x^3}} \, dx}{3003 b^4} \\ & = \frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(16 a) \int \frac {-\frac {819}{136} a b^3 (17 b c-8 a f)-\frac {495}{76} a b^3 (19 b d-10 a g) x-\frac {1001}{8} a b^4 e x^2}{\sqrt {a+b x^3}} \, dx}{15015 b^5} \\ & = \frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(16 a) \int \frac {-\frac {819}{136} a b^3 (17 b c-8 a f)-\frac {495}{76} a b^3 (19 b d-10 a g) x}{\sqrt {a+b x^3}} \, dx}{15015 b^5}-\frac {\left (2 a^2 e\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{15 b} \\ & = -\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}-\frac {\left (12 a^2 (19 b d-10 a g)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 b^{7/3}}-\frac {\left (6 a^2 \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{7/3}} \\ & = -\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}-\frac {24 a^2 (19 b d-10 a g) \sqrt {a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-10 a g) \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 )}{1729 b^{8/3} \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 (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 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 )}{1616615 b^{8/3} \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.91 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.23 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \sqrt {a+b x^3} \left (-\left (\left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \left (a (92378 e+90 x (988 f+935 g x))-3 b x \left (62985 c+11 x \left (4845 d+13 x \left (323 e+285 f x+255 g x^2\right )\right )\right )\right )\right )+11115 a (-17 b c+8 a f) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+8415 a (-19 b d+10 a g) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2078505 b^2 \sqrt {1+\frac {b x^3}{a}}} \]
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Time = 1.76 (sec) , antiderivative size = 956, normalized size of antiderivative = 1.30
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(956\) |
risch | \(\text {Expression too large to display}\) | \(1138\) |
default | \(\text {Expression too large to display}\) | \(1674\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.28 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2 \, {\left (93366 \, {\left (17 \, a^{2} b c - 8 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 100980 \, {\left (19 \, a^{2} b d - 10 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (765765 \, b^{3} g x^{8} + 855855 \, b^{3} f x^{7} + 969969 \, b^{3} e x^{6} + 323323 \, a b^{2} e x^{3} + 58905 \, {\left (19 \, b^{3} d + 3 \, a b^{2} g\right )} x^{5} + 77805 \, {\left (17 \, b^{3} c + 3 \, a b^{2} f\right )} x^{4} - 646646 \, a^{2} b e + 25245 \, {\left (19 \, a b^{2} d - 10 \, a^{2} b g\right )} x^{2} + 46683 \, {\left (17 \, a b^{2} c - 8 \, a^{2} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{14549535 \, b^{3}} \]
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Time = 2.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.32 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a} c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} f x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} g x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + e \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
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\[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \]
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\[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int x^3\,\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
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