Integrand size = 20, antiderivative size = 20 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=-\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}-\frac {\sqrt {c} (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \log ^{\frac {3}{2}}(f)}{2 e^4}+\frac {c \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{e^2}+\frac {(2 c d-b e)^2 \log ^2(f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{d+e x},x\right )}{2 e^4} \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(c \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}-\frac {((2 c d-b e) \log (f)) \int \frac {f^{a+b x+c x^2}}{(d+e x)^2} \, dx}{2 e^2} \\ & = -\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}+\frac {(c \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}-\frac {\left (c (2 c d-b e) \log ^2(f)\right ) \int f^{a+b x+c x^2} \, dx}{e^4}+\frac {\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4} \\ & = -\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}+\frac {(c \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac {\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4}-\frac {\left (c (2 c d-b e) f^{a-\frac {b^2}{4 c}} \log ^2(f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{e^4} \\ & = -\frac {f^{a+b x+c x^2}}{2 e (d+e x)^2}+\frac {(2 c d-b e) f^{a+b x+c x^2} \log (f)}{2 e^3 (d+e x)}-\frac {\sqrt {c} (2 c d-b e) f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \log ^{\frac {3}{2}}(f)}{2 e^4}+\frac {(c \log (f)) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{e^2}+\frac {\left ((2 c d-b e)^2 \log ^2(f)\right ) \int \frac {f^{a+b x+c x^2}}{d+e x} \, dx}{2 e^4} \\ \end{align*}
Not integrable
Time = 0.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {f^{c \,x^{2}+b x +a}}{\left (e x +d \right )^{3}}d x\]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Not integrable
Time = 0.74 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {f^{a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int { \frac {f^{c x^{2} + b x + a}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Not integrable
Time = 3.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {f^{a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {f^{c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^3} \,d x \]
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