Integrand size = 27, antiderivative size = 525 \[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\frac {g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}+\frac {\left (e (b d-a e) g^2 (1+m)+c \left (2 d^2 g^2 (1+p)+e^2 f^2 (3+m+2 p)-2 d e f g (3+m+2 p)\right )\right ) (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (1+m) (3+m+2 p)}-\frac {g (b e g (2+m+p)+2 c (d g (1+p)-e f (3+m+2 p))) (d+e x)^{2+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+m,-p,-p,3+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (2+m) (3+m+2 p)} \]
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Time = 0.41 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1667, 857, 773, 138} \[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (m+1,-p,-p,m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \left (g^2 (b d-a e)+\frac {c \left (2 d^2 g^2 (p+1)-2 d e f g (m+2 p+3)+e^2 f^2 (m+2 p+3)\right )}{e (m+1)}\right )}{c e^2 (m+2 p+3)}-\frac {g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} (b e g (m+p+2)+2 c d g (p+1)-2 c e f (m+2 p+3)) \operatorname {AppellF1}\left (m+2,-p,-p,m+3,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (m+2) (m+2 p+3)}+\frac {g^2 (d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)} \]
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Rule 138
Rule 773
Rule 857
Rule 1667
Rubi steps \begin{align*} \text {integral}& = \frac {g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}+\frac {\int (d+e x)^m \left (e \left (c e f^2 (3+m+2 p)-g^2 (a e (1+m)+b d (1+p))\right )-e g (2 c d g (1+p)+b e g (2+m+p)-2 c e f (3+m+2 p)) x\right ) \left (a+b x+c x^2\right )^p \, dx}{c e^2 (3+m+2 p)} \\ & = \frac {g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}-\frac {(g (2 c d g (1+p)+b e g (2+m+p)-2 c e f (3+m+2 p))) \int (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \, dx}{c e^2 (3+m+2 p)}+\frac {\left (e (b d-a e) g^2 (1+m)+c \left (2 d^2 g^2 (1+p)+e^2 f^2 (3+m+2 p)-2 d e f g (3+m+2 p)\right )\right ) \int (d+e x)^m \left (a+b x+c x^2\right )^p \, dx}{c e^2 (3+m+2 p)} \\ & = \frac {g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}-\frac {\left (g (2 c d g (1+p)+b e g (2+m+p)-2 c e f (3+m+2 p)) \left (a+b x+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p} \left (1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p}\right ) \text {Subst}\left (\int x^{1+m} \left (1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^p \, dx,x,d+e x\right )}{c e^3 (3+m+2 p)}+\frac {\left (\left (e (b d-a e) g^2 (1+m)+c \left (2 d^2 g^2 (1+p)+e^2 f^2 (3+m+2 p)-2 d e f g (3+m+2 p)\right )\right ) \left (a+b x+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p} \left (1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p}\right ) \text {Subst}\left (\int x^m \left (1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^p \, dx,x,d+e x\right )}{c e^3 (3+m+2 p)} \\ & = \frac {g^2 (d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}+\frac {\left (e (b d-a e) g^2 (1+m)+c \left (2 d^2 g^2 (1+p)+e^2 f^2 (3+m+2 p)-2 d e f g (3+m+2 p)\right )\right ) (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} F_1\left (1+m;-p,-p;2+m;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (1+m) (3+m+2 p)}-\frac {g (2 c d g (1+p)+b e g (2+m+p)-2 c e f (3+m+2 p)) (d+e x)^{2+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} F_1\left (2+m;-p,-p;3+m;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (2+m) (3+m+2 p)} \\ \end{align*}
\[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx \]
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\[\int \left (e x +d \right )^{m} \left (g x +f \right )^{2} \left (c \,x^{2}+b x +a \right )^{p}d x\]
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\[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (g x + f\right )}^{2} {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\text {Timed out} \]
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\[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (g x + f\right )}^{2} {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (g x + f\right )}^{2} {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx=\int {\left (f+g\,x\right )}^2\,{\left (d+e\,x\right )}^m\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \]
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