Integrand size = 35, antiderivative size = 714 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=-\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{7/3} e \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {b^2 (b c-6 a f) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
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Time = 0.78 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1839, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right )}{2240 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{224 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (b c-6 a f)}{24 a^{3/2}}+\frac {27 b^{7/3} e \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}-\frac {b \sqrt {a+b x^3} \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right )}{1680}-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right )}{2520} \]
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Rule 14
Rule 65
Rule 214
Rule 224
Rule 272
Rule 1839
Rule 1846
Rule 1849
Rule 1891
Rule 1892
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{9}-\frac {d x}{8}-\frac {e x^2}{7}-\frac {f x^3}{6}-\frac {g x^4}{5}\right )}{x^7} \, dx \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {1}{4} \left (27 b^2\right ) \int \frac {\frac {c}{54}+\frac {d x}{40}+\frac {e x^2}{28}+\frac {f x^3}{18}+\frac {g x^4}{10}}{x^4 \sqrt {a+b x^3}} \, dx \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (9 b^2\right ) \int \frac {-\frac {3 a d}{20}-\frac {3 a e x}{14}+\frac {1}{18} (b c-6 a f) x^2-\frac {3}{5} a g x^3}{x^3 \sqrt {a+b x^3}} \, dx}{8 a} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {\left (9 b^2\right ) \int \frac {\frac {6 a^2 e}{7}-\frac {2}{9} a (b c-6 a f) x-\frac {3}{20} a (b d-16 a g) x^2}{x^2 \sqrt {a+b x^3}} \, dx}{32 a^2} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (9 b^2\right ) \int \frac {\frac {4}{9} a^2 (b c-6 a f)+\frac {3}{10} a^2 (b d-16 a g) x-\frac {6}{7} a^2 b e x^2}{x \sqrt {a+b x^3}} \, dx}{64 a^3} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (9 b^2\right ) \int \frac {\frac {3}{10} a^2 (b d-16 a g)-\frac {6}{7} a^2 b e x}{\sqrt {a+b x^3}} \, dx}{64 a^3}-\frac {\left (b^2 (b c-6 a f)\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{16 a} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {\left (27 b^{8/3} e\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 (b^2 (b c-6 a f)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{48 a}-\frac {\left (27 b^2 \left (7 b d+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{4480 a} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{7/3} e \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} 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 )}{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 d+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{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 {(b (b c-6 a f)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{24 a} \\ & = -\frac {b \left (\frac {140 c}{x^6}+\frac {189 d}{x^5}+\frac {270 e}{x^4}+\frac {420 f}{x^3}+\frac {756 g}{x^2}\right ) \sqrt {a+b x^3}}{1680}-\frac {b^2 c \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 d \sqrt {a+b x^3}}{320 a x^2}-\frac {27 b^2 e \sqrt {a+b x^3}}{112 a x}+\frac {27 b^{7/3} e \sqrt {a+b x^3}}{112 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {280 c}{x^9}+\frac {315 d}{x^8}+\frac {360 e}{x^7}+\frac {420 f}{x^6}+\frac {504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {b^2 (b c-6 a f) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} 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 )}{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 d+20 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.70 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=-\frac {\sqrt {a+b x^3} \left (105 a^5 d \operatorname {Hypergeometric2F1}\left (-\frac {8}{3},-\frac {3}{2},-\frac {5}{3},-\frac {b x^3}{a}\right )+2 x \left (60 a^5 e \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},-\frac {3}{2},-\frac {4}{3},-\frac {b x^3}{a}\right )+7 x \left (5 a^3 f \left (a \left (2 a+5 b x^3\right ) \sqrt {1+\frac {b x^3}{a}}+3 b^2 x^6 \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )\right )+12 a^5 g x \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )-8 b^3 c x^6 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},1+\frac {b x^3}{a}\right )\right )\right )\right )}{840 a^4 x^8 \sqrt {1+\frac {b x^3}{a}}} \]
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Time = 1.83 (sec) , antiderivative size = 958, normalized size of antiderivative = 1.34
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(958\) |
risch | \(\text {Expression too large to display}\) | \(1160\) |
default | \(\text {Expression too large to display}\) | \(1273\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.26 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\left [-\frac {4860 \, a b^{\frac {5}{2}} e x^{9} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 210 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {a} x^{9} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (4860 \, a b^{2} e x^{8} + 6120 \, a^{2} b e x^{5} + 63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2880 \, a^{3} e x^{2} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}, -\frac {4860 \, a b^{\frac {5}{2}} e x^{9} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 420 \, {\left (b^{3} c - 6 \, a b^{2} f\right )} \sqrt {-a} x^{9} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 1701 \, {\left (a b^{2} d - 16 \, a^{2} b g\right )} \sqrt {b} x^{9} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (4860 \, a b^{2} e x^{8} + 6120 \, a^{2} b e x^{5} + 63 \, {\left (27 \, a b^{2} d + 208 \, a^{2} b g\right )} x^{7} + 840 \, {\left (a b^{2} c + 10 \, a^{2} b f\right )} x^{6} + 2880 \, a^{3} e x^{2} + 2520 \, a^{3} d x + 252 \, {\left (19 \, a^{2} b d + 16 \, a^{3} g\right )} x^{4} + 2240 \, a^{3} c + 560 \, {\left (7 \, a^{2} b c + 6 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{9}}\right ] \]
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Time = 13.50 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.80 \[ \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^{10}} \, dx=\frac {a^{\frac {3}{2}} d \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}} e \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}} g \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 {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 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 {\sqrt {a} b 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 {a^{2} c}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a^{2} f}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {11 a \sqrt {b} c}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} f}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {17 b^{\frac {3}{2}} c}{72 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} f}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {5}{2}} c}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} + \frac {b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{24 a^{\frac {3}{2}}} \]
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\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
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\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \]
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