Optimal. Leaf size=681 \[ \frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {24 a^2 e \sqrt {a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-728 a g+1547 b d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{85085 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a \sqrt {a+b x^3} (5 b c-2 a f)}{45 b^2}+\frac {6 a x \sqrt {a+b x^3} (17 b d-8 a g)}{935 b^2}+\frac {2 x^2 \sqrt {a+b x^3} \left (12155 c x+9945 d x^2+8415 e x^3+7293 f x^4+6435 g x^5\right )}{109395}+\frac {6 a e x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a f x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a g x^4 \sqrt {a+b x^3}}{187 b} \]
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Rubi [A] time = 1.42, antiderivative size = 681, normalized size of antiderivative = 1.00, 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 218
Rule 261
Rule 1594
Rule 1826
Rule 1836
Rule 1877
Rule 1878
Rule 1886
Rule 1888
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*}
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Mathematica [C] time = 0.30, 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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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{6} + f x^{5} + e x^{4} + d x^{3} + c x^{2}\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1197, normalized size = 1.76 \[ \text {result too large to display} \]
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 \[ \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} c}{9 \, b} + \int {\left (g x^{6} + f x^{5} + e x^{4} + d x^{3}\right )} \sqrt {b x^{3} + a}\,{d x} \]
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 \[ \int x^2\,\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.69, size = 223, normalized size = 0.33 \[ \frac {\sqrt {a} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} e x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} g x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + c \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) + f \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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