Optimal. Leaf size=746 \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (14 a f+b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{560 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a f+b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 b^{4/3} \sqrt {a+b x^3} (14 a f+b c)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {1}{420} \left (a+b x^3\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )-\frac {27 b \sqrt {a+b x^3} (14 a f+b c)}{112 a x}+\frac {27 b c \sqrt {a+b x^3}}{280 x^4}-\frac {b (4 a g+b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2} \]
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Rubi [A] time = 1.28, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (28 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (14 a f+b c)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{560 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a f+b c) E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 b^{4/3} \sqrt {a+b x^3} (14 a f+b c)}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{420} \left (a+b x^3\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right )-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {27 b \sqrt {a+b x^3} (14 a f+b c)}{112 a x}+\frac {27 b c \sqrt {a+b x^3}}{280 x^4}-\frac {b (4 a g+b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2} \]
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^8} \, dx &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}-\frac {g x^4}{3}\right )}{x^5} \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {1}{4} (27 a b) \int \frac {\frac {2 c}{35}+\frac {d x}{9}+\frac {2 e x^2}{5}-\frac {f x^3}{2}-\frac {2 g x^4}{9}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {1}{32} (27 b) \int \frac {-\frac {8 a d}{9}-\frac {16 a e x}{5}+\frac {2}{7} (b c+14 a f) x^2+\frac {16}{9} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {\frac {96 a^2 e}{5}-\frac {12}{7} a (b c+14 a f) x-\frac {8}{3} a (b d+4 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{64 a}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {(9 b) \int \frac {\frac {48}{7} a^2 (b c+14 a f)+\frac {32}{3} a^2 (b d+4 a g) x+\frac {96}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{256 a^2}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {-\frac {64}{3} a^3 (b d+4 a g)-\frac {192}{5} a^3 b e x-\frac {48}{7} a^2 b (b c+14 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{512 a^3}\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {(9 b) \int \frac {-\frac {192}{5} a^3 b e-\frac {48}{7} a^2 b (b c+14 a f) x}{\sqrt {a+b x^3}} \, dx}{512 a^3}+\frac {1}{8} (3 b (b d+4 a g)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}+\frac {\left (27 b^{5/3} (b c+14 a f)\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^{5/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1120}+\frac {1}{8} (b (b d+4 a g)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b c+14 a f) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{4/3} (b c+14 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 )}{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^{4/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\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 )}{560 \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 d+4 a g) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=\frac {27 b c \sqrt {a+b x^3}}{280 x^4}+\frac {b d \sqrt {a+b x^3}}{4 x^3}+\frac {27 b e \sqrt {a+b x^3}}{20 x^2}-\frac {27 b (b c+14 a f) \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{4/3} (b c+14 a f) \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}+\frac {140 g}{x^3}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (36 c x+70 d x^2+252 e x^3-315 f x^4-140 g x^5\right )}{140 x^5}-\frac {b (b d+4 a g) \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 c+14 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 )}{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^{4/3} \left (28 \sqrt [3]{b} e-\frac {5 \left (1-\sqrt {3}\right ) (b c+14 a f)}{a^{2/3}}\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 )}{560 \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 = 1.07, size = 240, normalized size = 0.32 \[ \frac {-\frac {60 a^2 c \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {7}{3},-\frac {3}{2};-\frac {4}{3};-\frac {b x^3}{a}\right )}{x^7}-\frac {84 a^2 e \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {5}{3},-\frac {3}{2};-\frac {2}{3};-\frac {b x^3}{a}\right )}{x^5}-\frac {105 a^2 f \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};-\frac {b x^3}{a}\right )}{x^4}+\frac {56 b g \left (a+b x^3\right )^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {b x^3}{a}+1\right )}{a^2}-105 b^2 d \sqrt {\frac {b x^3}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )-\frac {105 b d \left (a+b x^3\right )}{x^3}-\frac {70 d \left (a+b x^3\right )^2}{x^6}}{420 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1375, normalized size = 1.84 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.71, size = 536, normalized size = 0.72 \[ \frac {a^{\frac {3}{2}} c \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}} e \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}} f \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 {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 e \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 f \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 g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} - \frac {a^{2} d}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} d}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} g}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} d}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} g x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} d \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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