Integrand size = 20, antiderivative size = 139 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=-\frac {a \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (-\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1399, 524} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=-\frac {a \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (-\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a+b x^3+c x^6}\right ) \int \frac {\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{3/2}}{x^2} \, dx}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \\ & = -\frac {a \sqrt {a+b x^3+c x^6} F_1\left (-\frac {1}{3};-\frac {3}{2},-\frac {3}{2};\frac {2}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(379\) vs. \(2(139)=278\).
Time = 10.35 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\frac {10 \left (-80 a^2-61 a b x^3+19 b^2 x^6-70 a c x^6+29 b c x^9+10 c^2 x^{12}\right )+810 a b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+27 \left (b^2+20 a c\right ) x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{800 x \sqrt {a+b x^3+c x^6}} \]
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\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{2}}d x\]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^2} \,d x \]
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