Integrand size = 22, antiderivative size = 189 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \]
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Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {772, 138} \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{7/3} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \]
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Rule 138
Rule 772
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3}\right ) \text {Subst}\left (\int \frac {x^{17/3}}{\left (1-\frac {1}{2} \left (2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3} \left (1-\frac {1}{2} \left (2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{7/3}} \, dx,x,\frac {1}{d+e x}\right )}{16\ 2^{2/3} e \left (\frac {1}{d+e x}\right )^{14/3} \left (a+b x+c x^2\right )^{7/3}} \\ & = -\frac {3 \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{7/3} F_1\left (\frac {20}{3};\frac {7}{3},\frac {7}{3};\frac {23}{3};\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{320\ 2^{2/3} e (d+e x)^2 \left (a+b x+c x^2\right )^{7/3}} \\ \end{align*}
Time = 12.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 e^3 \sqrt [3]{\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \sqrt [3]{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \operatorname {AppellF1}\left (\frac {20}{3},\frac {7}{3},\frac {7}{3},\frac {23}{3},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{20\ 2^{2/3} c^2 (d+e x)^6 \sqrt [3]{a+x (b+c x)}} \]
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\[\int \frac {1}{\left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]
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Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]
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\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}} {\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]
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