Integrand size = 24, antiderivative size = 63 \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\frac {\left (a+b x+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)} \]
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Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {708, 272, 67} \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\frac {\left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d^3 (p+1) \left (b^2-4 a c\right )^2} \]
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Rule 67
Rule 272
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^p}{x^3} \, dx,x,b d+2 c d x\right )}{2 c d} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x}{4 c d^2}\right )^p}{x^2} \, dx,x,(b d+2 c d x)^2\right )}{4 c d} \\ & = \frac {(a+x (b+c x))^{1+p} \, _2F_1\left (2,1+p;2+p;1-\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\frac {(a+x (b+c x))^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 (1+p)} \]
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\[\int \frac {\left (c \,x^{2}+b x +a \right )^{p}}{\left (2 c d x +b d \right )^{3}}d x\]
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\[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (2 \, c d x + b d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\frac {\int \frac {\left (a + b x + c x^{2}\right )^{p}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (2 \, c d x + b d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (2 \, c d x + b d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^p}{(b d+2 c d x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (b\,d+2\,c\,d\,x\right )}^3} \,d x \]
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