Integrand size = 26, antiderivative size = 85 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {\left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1+m,-2 p,2+m,\frac {b (d+e x)}{b d-a e}\right )}{e (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {660, 72, 71} \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (m+1,-2 p,m+2,\frac {b (d+e x)}{b d-a e}\right )}{e (m+1)} \]
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Rule 71
Rule 72
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 \left (a b+b^2 x\right )^{2 p} (d+e x)^m \, dx \\ & = \left (\left (\frac {e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int (d+e x)^m \left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^{2 p} \, dx \\ & = \frac {\left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1+m,-2 p;2+m;\frac {b (d+e x)}{b d-a e}\right )}{e (1+m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {\left (\frac {e (a+b x)}{-b d+a e}\right )^{-2 p} \left ((a+b x)^2\right )^p (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1+m,-2 p,2+m,\frac {b (d+e x)}{b d-a e}\right )}{e (1+m)} \]
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\[\int \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}d x\]
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\[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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Exception generated. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \,d x \]
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