Integrand size = 22, antiderivative size = 180 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\frac {3 \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {8}{3},-\frac {4}{3},-\frac {4}{3},-\frac {5}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{\sqrt [3]{2} e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3}} \]
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Time = 0.14 (sec) , antiderivative size = 180, 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} \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\frac {3 \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {8}{3},-\frac {4}{3},-\frac {4}{3},-\frac {5}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{\sqrt [3]{2} e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}} \]
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Rule 138
Rule 772
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {1}{2} \left (2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{4/3} \left (1-\frac {1}{2} \left (2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}\right ) x\right )^{4/3}}{x^{11/3}} \, dx,x,\frac {1}{d+e x}\right )}{e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3}} \\ & = \frac {3 \left (a+b x+c x^2\right )^{4/3} F_1\left (-\frac {8}{3};-\frac {4}{3},-\frac {4}{3};-\frac {5}{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 )}{\sqrt [3]{2} e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3}} \\ \end{align*}
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx \]
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\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}}{e x +d}d x\]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {4}{3}}}{d + e x}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{d+e\,x} \,d x \]
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