Integrand size = 18, antiderivative size = 140 \[ \int x \sqrt {a+b x^3+c x^6} \, dx=\frac {x^2 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},-\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1399, 524} \[ \int x \sqrt {a+b x^3+c x^6} \, dx=\frac {x^2 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {1}{2},-\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x^3+c x^6} \int 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}}} \, 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 {x^2 \sqrt {a+b x^3+c x^6} F_1\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 )}{2 \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. \(337\) vs. \(2(140)=280\).
Time = 10.03 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.41 \[ \int x \sqrt {a+b x^3+c x^6} \, dx=\frac {x^2 \left (10 \left (a+b x^3+c x^6\right )+15 a \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+3 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{50 \sqrt {a+b x^3+c x^6}} \]
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\[\int x \sqrt {c \,x^{6}+b \,x^{3}+a}d x\]
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\[ \int x \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} x \,d x } \]
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\[ \int x \sqrt {a+b x^3+c x^6} \, dx=\int x \sqrt {a + b x^{3} + c x^{6}}\, dx \]
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\[ \int x \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} x \,d x } \]
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\[ \int x \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} x \,d x } \]
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Timed out. \[ \int x \sqrt {a+b x^3+c x^6} \, dx=\int x\,\sqrt {c\,x^6+b\,x^3+a} \,d x \]
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