Integrand size = 26, antiderivative size = 71 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,1+2 p,2 (1+p),-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e) (1+2 p)} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {660, 70} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,2 p+1,2 (p+1),-\frac {e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)} \]
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Rule 70
Rule 660
Rubi steps \begin{align*} \text {integral}& = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (a b+b^2 x\right )^{2 p}}{d+e x} \, dx \\ & = \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,1+2 p;2 (1+p);-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e) (1+2 p)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=-\frac {(a+b x) \left ((a+b x)^2\right )^p \operatorname {Hypergeometric2F1}\left (1,1+2 p,2+2 p,\frac {e (a+b x)}{-b d+a e}\right )}{(-b d+a e) (1+2 p)} \]
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\[\int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{e x +d}d x\]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{p}}{d + e x}\, dx \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p}{d+e\,x} \,d x \]
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