Optimal. Leaf size=692 \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{40 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{16 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )-\frac {b (4 a f+b c) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x} \]
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Rubi [A] time = 1.00, antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {14, 1825, 1826, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{40 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{16 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {b (4 a f+b c) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 208
Rule 218
Rule 266
Rule 1825
Rule 1826
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^7} \, dx &=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{6}-\frac {d x}{5}-\frac {e x^2}{4}-\frac {f x^3}{3}-\frac {g x^4}{2}\right )}{x^4} \, dx\\ &=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {1}{4} (27 a b) \int \frac {\frac {c}{9}+\frac {2 d x}{5}-\frac {e x^2}{2}-\frac {2 f x^3}{9}-\frac {g x^4}{5}}{x^4 \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {1}{8} (9 b) \int \frac {-\frac {12 a d}{5}+3 a e x+\frac {1}{3} (b c+4 a f) x^2+\frac {6}{5} a g x^3}{x^3 \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {(9 b) \int \frac {-12 a^2 e-\frac {4}{3} a (b c+4 a f) x-\frac {12}{5} a (b d+2 a g) x^2}{x^2 \sqrt {a+b x^3}} \, dx}{32 a}\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {(9 b) \int \frac {\frac {8}{3} a^2 (b c+4 a f)+\frac {24}{5} a^2 (b d+2 a g) x+12 a^2 b e x^2}{x \sqrt {a+b x^3}} \, dx}{64 a^2}\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {(9 b) \int \frac {\frac {24}{5} a^2 (b d+2 a g)+12 a^2 b e x}{\sqrt {a+b x^3}} \, dx}{64 a^2}+\frac {1}{8} (3 b (b c+4 a f)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {1}{16} \left (27 b^{5/3} e\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx+\frac {1}{8} (b (b c+4 a f)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {1}{80} \left (27 b \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} 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 )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+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 )}{40 \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 c+4 a f) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {b (b c+4 a f) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} 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 )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+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 )}{40 \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 = 0.73, size = 240, normalized size = 0.35 \[ \frac {-\frac {12 a^2 d \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {5}{3},-\frac {3}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )}{x^5}-\frac {15 a^2 e \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};-\frac {b x^3}{a}\right )}{x^4}+\frac {8 b f \left (a+b x^3\right )^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^3}{a}+1\right )}{a^2}-\frac {30 a^2 g \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {3}{2},-\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{x^2}-15 b^2 c \sqrt {\frac {b x^3}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )-\frac {15 b c \left (a+b x^3\right )}{x^3}-\frac {10 c \left (a+b x^3\right )^2}{x^6}}{60 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1196, normalized size = 1.73 \[ \text {result too large to display} \]
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 \[ \frac {1}{24} \, {\left (\frac {3 \, b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, {\left (5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {b x^{3} + a} a b^{2}\right )}}{{\left (b x^{3} + a\right )}^{2} - 2 \, {\left (b x^{3} + a\right )} a + a^{2}}\right )} c + \int \frac {{\left (b g x^{6} + b f x^{5} + b e x^{4} + a f x^{2} + {\left (b d + a g\right )} x^{3} + a e x + a d\right )} \sqrt {b x^{3} + a}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.92, size = 524, normalized size = 0.76 \[ \frac {a^{\frac {3}{2}} d \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}} e \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 {a^{\frac {3}{2}} g \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 d \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 e \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 )} - \sqrt {a} b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b g 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^{2} c}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} c}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} f}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} c}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} f x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} c \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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