Optimal. Leaf size=231 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} (g (m+n+2) (a e g (m+n+3)-c d (d g (n+1)+e f (m+2)))+c (m+2) (e f-d g) (d g (n+1)+e f (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{e^2 g^2 (m+1) (m+n+2) (m+n+3)}-\frac{c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \]
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Rubi [A] time = 0.261532, antiderivative size = 227, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {951, 80, 70, 69} \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \left (a e g (m+n+3)+\frac{c (m+2) (e f-d g) (d g (n+1)+e f (m+1))}{g (m+n+2)}-c d (d g (n+1)+e f (m+2))\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{e^2 g (m+1) (m+n+3)}-\frac{c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \]
Antiderivative was successfully verified.
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Rule 951
Rule 80
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\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*}
Mathematica [A] time = 0.17647, size = 179, normalized size = 0.77 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \left (e \left (a g^2+c f (e f-2 d g)\right ) \, _2F_1\left (m+1,-n;m+2;\frac{g (d+e x)}{d g-e f}\right )+c (e f-d g)^2 \, _2F_1\left (m+1,-n-2;m+2;\frac{g (d+e x)}{d g-e f}\right )-2 c (e f-d g)^2 \, _2F_1\left (m+1,-n-1;m+2;\frac{g (d+e x)}{d g-e f}\right )\right )}{e^2 g^2 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.683, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{n} \left ( ce{x}^{2}+2\,cdx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c e x^{2} + 2 \, c d x + a\right )}{\left (e x + d\right )}^{m}{\left (g x + f\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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