Integrand size = 28, antiderivative size = 231 \[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=-\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {(c (e f-d g) (2+m) (e f (1+m)+d g (1+n))+g (2+m+n) (a e g (3+m+n)-c d (e f (2+m)+d g (1+n)))) (d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g^2 (1+m) (2+m+n) (3+m+n)} \]
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Time = 0.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {965, 81, 72, 71} \[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (a e g (m+n+3)+\frac {c (m+2) (e f-d g) (d g (n+1)+e f (m+1))}{g (m+n+2)}-c d (d g (n+1)+e f (m+2))\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g (m+1) (m+n+3)}-\frac {c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \]
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Rule 71
Rule 72
Rule 81
Rule 965
Rubi steps \begin{align*} \text {integral}& = \frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\int (d+e x)^m (f+g x)^n \left (e (a e g (3+m+n)-c d (e f (2+m)+d g (1+n)))-c e^2 (e f-d g) (2+m) x\right ) \, dx}{e^2 g (3+m+n)} \\ & = -\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) \int (d+e x)^m (f+g x)^n \, dx}{e g (3+m+n)} \\ & = -\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n}\right ) \int (d+e x)^m \left (\frac {e f}{e f-d g}+\frac {e g x}{e f-d g}\right )^n \, dx}{e g (3+m+n)} \\ & = -\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) (d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g (1+m) (3+m+n)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.77 \[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\frac {(d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (c (e f-d g)^2 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )-2 c (e f-d g)^2 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )+e \left (a g^2+c f (e f-2 d g)\right ) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )\right )}{e^2 g^2 (1+m)} \]
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\[\int \left (e x +d \right )^{m} \left (g x +f \right )^{n} \left (c e \,x^{2}+2 c d x +a \right )d x\]
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\[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\int { {\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n} \,d x } \]
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Exception generated. \[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\int { {\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n} \,d x } \]
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\[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\int { {\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx=\int {\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^m\,\left (c\,e\,x^2+2\,c\,d\,x+a\right ) \,d x \]
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