Optimal. Leaf size=667 \[ -\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5005 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 b c-4 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {6 a \sqrt {a+b x^3} (13 b c-4 a f)}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a \sqrt {a+b x^3} (5 b d-2 a g)}{45 b^2}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b} \]
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Rubi [A] time = 1.05, antiderivative size = 667, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1826, 1836, 1888, 1886, 261, 1878, 218, 1877} \[ -\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5005 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 b c-4 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {6 a \sqrt {a+b x^3} (13 b c-4 a f)}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a \sqrt {a+b x^3} (5 b d-2 a g)}{45 b^2}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b} \]
Antiderivative was successfully verified.
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Rule 218
Rule 261
Rule 1826
Rule 1836
Rule 1877
Rule 1878
Rule 1886
Rule 1888
Rubi steps
\begin {align*} \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {1}{2} (3 a) \int \frac {x \left (\frac {2 c}{7}+\frac {2 d x}{9}+\frac {2 e x^2}{11}+\frac {2 f x^3}{13}+\frac {2 g x^4}{15}\right )}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {a \int \frac {x \left (\frac {9 b c}{7}+\frac {1}{5} (5 b d-2 a g) x+\frac {9}{11} b e x^2+\frac {9}{13} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{3 b}\\ &=\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(2 a) \int \frac {x \left (\frac {9}{26} b (13 b c-4 a f)+\frac {7}{10} b (5 b d-2 a g) x+\frac {63}{22} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{21 b^2}\\ &=\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(4 a) \int \frac {-\frac {63}{22} a b^2 e+\frac {45}{52} b^2 (13 b c-4 a f) x+\frac {7}{4} b^2 (5 b d-2 a g) x^2}{\sqrt {a+b x^3}} \, dx}{105 b^3}\\ &=\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(4 a) \int \frac {-\frac {63}{22} a b^2 e+\frac {45}{52} b^2 (13 b c-4 a f) x}{\sqrt {a+b x^3}} \, dx}{105 b^3}+\frac {(a (5 b d-2 a g)) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{15 b}\\ &=\frac {2 a (5 b d-2 a g) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(3 a (13 b c-4 a f)) \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 (3 a^{4/3} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{5005 b^{4/3}}\\ &=\frac {2 a (5 b d-2 a g) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a (13 b c-4 a f) \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 \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} (13 b c-4 a f) \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 {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\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 )}{5005 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}}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 143, normalized size = 0.21 \[ \frac {\sqrt {a+b x^3} \left (495 b x^2 (13 b c-4 a f) \, _2F_1\left (-\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )-4 \left (a+b x^3\right ) \sqrt {\frac {b x^3}{a}+1} \left (286 a g-b \left (715 d+585 e x+495 f x^2+429 g x^3\right )\right )-2340 a b e x \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )\right )}{12870 b^2 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{5} + f x^{4} + e x^{3} + d x^{2} + c x\right )} \sqrt {b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1311, normalized size = 1.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.39, size = 223, normalized size = 0.33 \[ \frac {\sqrt {a} c x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\sqrt {a} e x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} f x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + d \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) + g \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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