Integrand size = 30, antiderivative size = 377 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}} \]
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Time = 0.25 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1839, 1846, 272, 65, 214, 1899, 281, 223, 212, 1212, 226, 1210} \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {a} f+5 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {1}{2} b^{3/2} e \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{560} b \sqrt {a+b x^4} \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right )-\frac {1}{840} \left (a+b x^4\right )^{3/2} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \]
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Rule 14
Rule 65
Rule 212
Rule 214
Rule 223
Rule 226
Rule 272
Rule 281
Rule 1210
Rule 1212
Rule 1839
Rule 1846
Rule 1899
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{8}-\frac {d x}{7}-\frac {e x^2}{6}-\frac {f x^3}{5}\right ) \sqrt {a+b x^4}}{x^5} \, dx \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {c}{32}+\frac {d x}{21}+\frac {e x^2}{12}+\frac {f x^3}{5}}{x \sqrt {a+b x^4}} \, dx \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {d}{21}+\frac {e x}{12}+\frac {f x^2}{5}}{\sqrt {a+b x^4}} \, dx+\frac {1}{8} \left (3 b^2 c\right ) \int \frac {1}{x \sqrt {a+b x^4}} \, dx \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \left (\frac {e x}{12 \sqrt {a+b x^4}}+\frac {\frac {d}{21}+\frac {f x^2}{5}}{\sqrt {a+b x^4}}\right ) \, dx+\frac {1}{32} \left (3 b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right ) \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {d}{21}+\frac {f x^2}{5}}{\sqrt {a+b x^4}} \, dx+\frac {1}{16} (3 b c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )+\left (b^2 e\right ) \int \frac {x}{\sqrt {a+b x^4}} \, dx \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {1}{2} \left (b^2 e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {1}{5} \left (12 \sqrt {a} b^{3/2} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{35} \left (4 b^2 \left (5 d+\frac {21 \sqrt {a} f}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\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}\right ) \left (a+b x^4\right )^{3/2}-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} \left (b^2 e\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right ) \\ & = -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\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}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.46 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=-\frac {\sqrt {a+b x^4} \left (240 a^2 d x \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+7 \left (15 c \left (a \left (2 a+5 b x^4\right ) \sqrt {1+\frac {b x^4}{a}}+3 b^2 x^8 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )\right )+40 a^2 e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b x^4}{a}\right )+48 a^2 f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{4},-\frac {1}{4},-\frac {b x^4}{a}\right )\right )\right )}{1680 a x^8 \sqrt {1+\frac {b x^4}{a}}} \]
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Result contains complex when optimal does not.
Time = 2.25 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (2352 b f \,x^{7}+1120 b e \,x^{6}+720 b d \,x^{5}+525 b c \,x^{4}+336 a f \,x^{3}+280 a e \,x^{2}+240 a d x +210 a c \right )}{1680 x^{8}}+\frac {b^{2} \left (\frac {160 d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {672 i f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {140 e \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{\sqrt {b}}-\frac {105 c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}\right )}{280}\) | \(295\) |
elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a d \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a e \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {a f \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {5 b c \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b d \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b e \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {7 b f \sqrt {b \,x^{4}+a}}{5 x}+\frac {4 b^{2} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} e \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {12 i b^{\frac {3}{2}} f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} c \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(351\) |
default | \(e \left (\frac {b^{\frac {3}{2}} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(357\) |
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\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{9}} \,d x } \]
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Result contains complex when optimal does not.
Time = 6.50 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b e}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {a^{2} c}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 a \sqrt {b} c}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} c}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} - \frac {3 b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} - \frac {b^{2} e x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
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\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{9}} \,d x } \]
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\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^9} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^9} \,d x \]
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