Integrand size = 16, antiderivative size = 136 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.03 (sec) , antiderivative size = 136, 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} \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 440
Rule 1362
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a+b x^3+c x^6}\right ) \int \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} \, 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 x \sqrt {a+b x^3+c x^6} F_1\left (\frac {1}{3};-\frac {3}{2},-\frac {3}{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. \(408\) vs. \(2(136)=272\).
Time = 10.48 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.00 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {x \left (8 \left (27 a b^2+364 a^2 c+27 b^3 x^3+548 a b c x^3+211 b^2 c x^6+476 a c^2 x^6+296 b c^2 x^9+112 c^3 x^{12}\right )-216 a \left (b^2-28 a c\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )-27 b \left (5 b^2-44 a c\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{8960 c \sqrt {a+b x^3+c x^6}} \]
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\[\int \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}d x\]
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\[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\int {\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \]
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