Integrand size = 21, antiderivative size = 139 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^3 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{3/2}}-\frac {45 b^2 c^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{9},\frac {1}{2},\frac {11}{9},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}} \]
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Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {376, 348, 283, 331, 372, 371} \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=-\frac {45 b^2 c^3 x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {2}{9},\frac {1}{2},\frac {11}{9},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8} \]
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Rule 283
Rule 331
Rule 348
Rule 371
Rule 372
Rule 376
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x^9} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (2 \text {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^9}}{x^{17}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}+\text {Subst}\left (\frac {1}{16} \left (9 b c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\text {Subst}\left (\frac {\left (45 b^2 c^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right )}{224 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\text {Subst}\left (\frac {\left (45 b^2 c^3 \sqrt {1+\frac {b c^{3/2} x^{9/2}}{a}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+\frac {b c^{3/2} x^9}{a}}} \, dx,x,\sqrt {x}\right )}{224 a \sqrt {a+b c^{3/2} x^{9/2}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {45 b^2 c^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}} \\ \end{align*}
\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx \]
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\[\int \frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{x^{9}}d x\]
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\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int { \frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a}}{x^{9}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int \frac {\sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x^{9}}\, dx \]
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\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int { \frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a}}{x^{9}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int { \frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x^9} \,d x \]
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