Optimal. Leaf size=705 \[ -\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (14 a g+b d)\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a g+b d) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 b^{4/3} \sqrt {a+b x^3} (14 a g+b d)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {1}{560} b \sqrt {a+b x^3} \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right )-\frac {1}{840} \left (a+b x^3\right )^{3/2} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \]
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Rubi [A] time = 1.01, antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ -\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (14 a g+b d)\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a g+b d) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^{4/3} \sqrt {a+b x^3} (14 a g+b d)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {1}{560} b \sqrt {a+b x^3} \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right )-\frac {1}{840} \left (a+b x^3\right )^{3/2} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 208
Rule 218
Rule 266
Rule 1825
Rule 1832
Rule 1835
Rule 1877
Rule 1878
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^9} \, dx &=-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{8}-\frac {d x}{7}-\frac {e x^2}{6}-\frac {f x^3}{5}-\frac {g x^4}{4}\right )}{x^6} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{4} \left (27 b^2\right ) \int \frac {\frac {c}{40}+\frac {d x}{28}+\frac {e x^2}{18}+\frac {f x^3}{10}+\frac {g x^4}{4}}{x^3 \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {\left (27 b^2\right ) \int \frac {-\frac {a d}{7}-\frac {2 a e x}{9}+\frac {1}{40} (b c-16 a f) x^2-a g x^3}{x^2 \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {\left (27 b^2\right ) \int \frac {\frac {4 a^2 e}{9}-\frac {1}{20} a (b c-16 a f) x+\frac {1}{7} a (b d+14 a g) x^2}{x \sqrt {a+b x^3}} \, dx}{32 a^2}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {\left (27 b^2\right ) \int \frac {-\frac {1}{20} a (b c-16 a f)+\frac {1}{7} a (b d+14 a g) x}{\sqrt {a+b x^3}} \, dx}{32 a^2}+\frac {1}{8} \left (3 b^2 e\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac {1}{8} \left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {\left (27 b^{5/3} (b d+14 a g)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{224 a}-\frac {\left (27 b^2 \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{4480 a}\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b d+14 a g) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b d+14 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 )}{224 a^{2/3} \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 {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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 )}{2240 a \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 {1}{4} (b e) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=-\frac {1}{560} b \left (\frac {63 c}{x^5}+\frac {90 d}{x^4}+\frac {140 e}{x^3}+\frac {252 f}{x^2}+\frac {630 g}{x}\right ) \sqrt {a+b x^3}-\frac {27 b^2 c \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 d \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b d+14 a g) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}+\frac {210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac {b^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b d+14 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 )}{224 a^{2/3} \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 {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (7 (b c-16 a f)+\frac {20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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 )}{2240 a \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 = 0.60, size = 202, normalized size = 0.29 \[ -\frac {\sqrt {a+b x^3} \left (2 x \left (7 x \left (5 \left (3 a^2 g x^2 \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};-\frac {b x^3}{a}\right )+3 b^2 e x^6 \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )+a e \left (2 a+5 b x^3\right ) \sqrt {\frac {b x^3}{a}+1}\right )+12 a^2 f x \, _2F_1\left (-\frac {5}{3},-\frac {3}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )\right )+60 a^2 d \, _2F_1\left (-\frac {7}{3},-\frac {3}{2};-\frac {4}{3};-\frac {b x^3}{a}\right )\right )+105 a^2 c \, _2F_1\left (-\frac {8}{3},-\frac {3}{2};-\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{840 a x^8 \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{9}}, x\right ) \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1663, normalized size = 2.36 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{9}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.28, size = 527, normalized size = 0.75 \[ \frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {a^{\frac {3}{2}} g \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {\sqrt {a} b g \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {a^{2} e}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} e}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} e}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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