Integrand size = 20, antiderivative size = 140 \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {a+b x^3+c x^6}} \]
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Time = 0.08 (sec) , antiderivative size = 140, 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} \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {a+b x^3+c x^6}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\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}}}\right ) \int \frac {1}{x^3 \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}}}} \, dx}{\sqrt {a+b x^3+c x^6}} \\ & = -\frac {\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}}} F_1\left (-\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {1}{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 )}{2 x^2 \sqrt {a+b x^3+c x^6}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(140)=280\).
Time = 10.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\frac {-4 \left (a+b x^3+c x^6\right )-2 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+c x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{8 a x^2 \sqrt {a+b x^3+c x^6}} \]
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\[\int \frac {1}{x^{3} \sqrt {c \,x^{6}+b \,x^{3}+a}}d x\]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x^{3} + c x^{6}}}\, dx \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int { \frac {1}{\sqrt {c x^{6} + b x^{3} + a} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {1}{x^3\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \]
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