Integrand size = 26, antiderivative size = 340 \[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=-\frac {d (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {(d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \]
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Time = 0.43 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1403, 1804, 1362, 440, 1399, 524} \[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\frac {(d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \]
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Rule 440
Rule 524
Rule 1362
Rule 1399
Rule 1403
Rule 1804
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-d+x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {d}{\sqrt {a+b x^3+c x^6}}+\frac {x}{\sqrt {a+b x^3+c x^6}}\right ) \, dx,x,d+e x\right )}{e^2} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2} \\ & = \frac {\left (\sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \text {Subst}\left (\int \frac {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,x,d+e x\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {\left (d \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \text {Subst}\left (\int \frac {1}{\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,x,d+e x\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \\ & = -\frac {d (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {(d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \\ \end{align*}
\[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx \]
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\[\int \frac {x}{\sqrt {a +b \left (e x +d \right )^{3}+c \left (e x +d \right )^{6}}}d x\]
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\[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x}{\sqrt {a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]
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\[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x}{\sqrt {a+b\,{\left (d+e\,x\right )}^3+c\,{\left (d+e\,x\right )}^6}} \,d x \]
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