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 = 0.35, antiderivative size = 263, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 69
Rule 70
Rule 80
Rule 951
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*}
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Mathematica [A] time = 0.22, 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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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+b x +a \right ) \left (e x +d \right )^{m} \left (g x +f \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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