Optimal. Leaf size=723 \[ -\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b c-4 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} (19 b c-4 a f)}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a^2 \sqrt {a+b x^3} (7 b d-2 a g)}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845} \]
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Rubi [A] time = 1.24, antiderivative size = 723, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} (19 b c-4 a f)}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b c-4 a f) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a^2 \sqrt {a+b x^3} (7 b d-2 a g)}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845} \]
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 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {1}{2} (9 a) \int x \sqrt {a+b x^3} \left (\frac {2 c}{13}+\frac {2 d x}{15}+\frac {2 e x^2}{17}+\frac {2 f x^3}{19}+\frac {2 g x^4}{21}\right ) \, dx\\ &=\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {1}{4} \left (27 a^2\right ) \int \frac {x \left (\frac {4 c}{91}+\frac {4 d x}{135}+\frac {4 e x^2}{187}+\frac {4 f x^3}{247}+\frac {4 g x^4}{315}\right )}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (3 a^2\right ) \int \frac {x \left (\frac {18 b c}{91}+\frac {2}{105} (7 b d-2 a g) x+\frac {18}{187} b e x^2+\frac {18}{247} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{2 b}\\ &=\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (3 a^2\right ) \int \frac {x \left (\frac {9}{247} b (19 b c-4 a f)+\frac {1}{15} b (7 b d-2 a g) x+\frac {63}{187} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{7 b^2}\\ &=\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (6 a^2\right ) \int \frac {-\frac {63}{187} a b^2 e+\frac {45}{494} b^2 (19 b c-4 a f) x+\frac {1}{6} b^2 (7 b d-2 a g) x^2}{\sqrt {a+b x^3}} \, dx}{35 b^3}\\ &=\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (6 a^2\right ) \int \frac {-\frac {63}{187} a b^2 e+\frac {45}{494} b^2 (19 b c-4 a f) x}{\sqrt {a+b x^3}} \, dx}{35 b^3}+\frac {\left (a^2 (7 b d-2 a g)\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{35 b}\\ &=\frac {2 a^2 (7 b d-2 a g) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (27 a^2 (19 b c-4 a f)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 b^{4/3}}-\frac {\left (27 a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{4/3}}\\ &=\frac {2 a^2 (7 b d-2 a g) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 (19 b c-4 a f) \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 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 )}{1729 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 {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 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 )}{1616615 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 = 0.37, size = 148, normalized size = 0.20 \[ -\frac {\sqrt {a+b x^3} \left (7980 a^2 b e x \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+1785 a b x^2 (4 a f-19 b c) \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )+4 \left (a+b x^3\right )^2 \sqrt {\frac {b x^3}{a}+1} (646 a g-2261 b d-5 b x (399 e+17 x (21 f+19 g x)))\right )}{67830 b^2 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b g x^{8} + b f x^{7} + b e x^{6} + {\left (b d + a g\right )} x^{5} + a e x^{3} + {\left (b c + a f\right )} x^{4} + a d x^{2} + a c x\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1383, normalized size = 1.91 \[ \text {result too large to display} \]
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 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x\,{d x} \]
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 \[ \int x\,{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.78, size = 525, normalized size = 0.73 \[ \frac {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 )} + \frac {\sqrt {a} b c 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} b e 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} b f 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 )} + a 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 ) + a 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 ) + b d \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 ) + b g \left (\begin {cases} \frac {16 a^{3} \sqrt {a + b x^{3}}}{315 b^{3}} - \frac {8 a^{2} x^{3} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 a x^{6} \sqrt {a + b x^{3}}}{105 b} + \frac {2 x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{9}}{9} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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