Optimal. Leaf size=694 \[ -\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-4 a g) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} (19 b d-4 a g)}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835} \]
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Rubi [A] time = 0.90, antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1853, 1888, 1886, 261, 1878, 218, 1877} \[ \frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 \sqrt {a+b x^3} (19 b d-4 a g)}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-4 a g) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {2 \left (a+b x^3\right )^{3/2} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835} \]
Antiderivative was successfully verified.
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Rule 218
Rule 261
Rule 1853
Rule 1877
Rule 1878
Rule 1886
Rule 1888
Rubi steps
\begin {align*} \int \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 \left (a+b x^3\right )^{3/2} \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} (9 a) \int \sqrt {a+b 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 ) \, dx\\ &=\frac {2 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {1}{4} \left (27 a^2\right ) \int \frac {\frac {4 c}{55}+\frac {4 d x}{91}+\frac {4 e x^2}{135}+\frac {4 f x^3}{187}+\frac {4 g x^4}{247}}{\sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {14 b c}{55}+\frac {2}{247} (19 b d-4 a g) x+\frac {14}{135} b e x^2+\frac {14}{187} b f x^3}{\sqrt {a+b x^3}} \, dx}{14 b}\\ &=\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {7}{187} b (17 b c-2 a f)+\frac {5}{247} b (19 b d-4 a g) x+\frac {7}{27} b^2 e x^2}{\sqrt {a+b x^3}} \, dx}{35 b^2}\\ &=\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2\right ) \int \frac {\frac {7}{187} b (17 b c-2 a f)+\frac {5}{247} b (19 b d-4 a g) x}{\sqrt {a+b x^3}} \, dx}{35 b^2}+\frac {1}{5} \left (a^2 e\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}+\frac {\left (27 a^2 (19 b d-4 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^{4/3}}+\frac {\left (27 a^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{4/3}}\\ &=\frac {2 a^2 e \sqrt {a+b x^3}}{15 b}+\frac {54 a^2 f x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 g x^2 \sqrt {a+b x^3}}{1729 b}+\frac {54 a^2 (19 b d-4 a g) \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 \left (a+b x^3\right )^{3/2} \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 \sqrt {a+b x^3} \left (793611 c x+479655 d x^2+323323 e x^3+233415 f x^4+176715 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-4 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^{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^2 \left (1729 \sqrt [3]{b} (17 b c-2 a f)-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-4 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^{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 = 0.26, size = 139, normalized size = 0.20 \[ \frac {\sqrt {a+b x^3} \left (-570 a x (2 a f-17 b c) \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-255 a x^2 (4 a g-19 b d) \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )+4 \left (a+b x^3\right )^2 \sqrt {\frac {b x^3}{a}+1} (323 e+15 x (19 f+17 g x))\right )}{9690 b \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}, x\right ) \]
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 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1629, normalized size = 2.35 \[ \text {result too large to display} \]
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 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.13, size = 444, normalized size = 0.64 \[ \frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} d 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}} f 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}} g 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^{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} b 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} b 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} b 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 )} + a e \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 ) + b 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 ) \]
Verification of antiderivative is not currently implemented for this CAS.
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