Optimal. Leaf size=265 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )-(d g (n+1)+e f (m+1)) (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}+\frac{(d+e x)^{m+1} (f+g x)^{n+1} (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]
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Rubi [A] time = 0.350857, antiderivative size = 263, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {951, 80, 70, 69} \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )+(d g (n+1)+e f (m+1)) (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}-\frac{(d+e x)^{m+1} (f+g x)^{n+1} (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]
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+b x+c x^2\right ) \, dx &=\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\int (d+e x)^m (f+g x)^n \left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))-e (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) x\right ) \, dx}{e^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+n)}\right ) \int (d+e x)^m (f+g x)^n \, dx}{e^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+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^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+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^3 g (1+m) (3+m+n)}\\ \end{align*}
Mathematica [A] time = 0.208153, size = 187, normalized size = 0.71 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \left (e \left (e \left (g (a g-b f)+c f^2\right ) \, _2F_1\left (m+1,-n;m+2;\frac{g (d+e x)}{d g-e f}\right )-(2 c f-b g) (e f-d g) \, _2F_1\left (m+1,-n-1;m+2;\frac{g (d+e x)}{d g-e f}\right )\right )+c (e f-d g)^2 \, _2F_1\left (m+1,-n-2;m+2;\frac{g (d+e x)}{d g-e f}\right )\right )}{e^3 g^2 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.623, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{n} \left ( c{x}^{2}+bx+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 x^{2} + b 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 x^{2} + b 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 x^{2} + b 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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