Integrand size = 16, antiderivative size = 135 \[ \int \sqrt {a+b x^3+c x^6} \, dx=\frac {x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1362, 440} \[ \int \sqrt {a+b x^3+c x^6} \, dx=\frac {x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 440
Rule 1362
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x^3+c x^6} \int \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 \sqrt {a+b x^3+c x^6} F_1\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 )}{\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. \(335\) vs. \(2(135)=270\).
Time = 10.24 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.48 \[ \int \sqrt {a+b x^3+c x^6} \, dx=\frac {x \left (8 \left (a+b x^3+c x^6\right )+24 a \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+3 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{32 \sqrt {a+b x^3+c x^6}} \]
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\[\int \sqrt {c \,x^{6}+b \,x^{3}+a}d x\]
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\[ \int \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} \,d x } \]
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\[ \int \sqrt {a+b x^3+c x^6} \, dx=\int \sqrt {a + b x^{3} + c x^{6}}\, dx \]
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\[ \int \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} \,d x } \]
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\[ \int \sqrt {a+b x^3+c x^6} \, dx=\int { \sqrt {c x^{6} + b x^{3} + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b x^3+c x^6} \, dx=\int \sqrt {c\,x^6+b\,x^3+a} \,d x \]
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