Integrand size = 20, antiderivative size = 143 \[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x^4 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {3}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 a \sqrt {a+b x^3+c x^6}} \]
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Time = 0.08 (sec) , antiderivative size = 143, 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} \[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x^4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {3}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 a \sqrt {a+b x^3+c x^6}} \]
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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 {x^3}{\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}{a \sqrt {a+b x^3+c x^6}} \\ & = \frac {x^4 \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 {4}{3};\frac {3}{2},\frac {3}{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 )}{4 a \sqrt {a+b x^3+c x^6}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(340\) vs. \(2(143)=286\).
Time = 10.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.38 \[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x \left (-2 \left (b+2 c x^3\right )+2 b \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+c x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \]
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\[\int \frac {x^{3}}{\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \]
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