Optimal. Leaf size=263 \[ \frac{(a+b x)^{m+1} (B e-A f) (c+d x)^{n+1} (e+f x)^{-m-n-2}}{(m+n+2) (b e-a f) (d e-c f)}-\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} (a (A d f (m+1)+B (d e (n+1)-c f (m+n+2)))+b (A (c f (n+1)-d e (m+n+2))+B c e (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)} \]
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Rubi [A] time = 0.226597, antiderivative size = 261, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {155, 12, 132} \[ \frac{(a+b x)^{m+1} (B e-A f) (c+d x)^{n+1} (e+f x)^{-m-n-2}}{(m+n+2) (b e-a f) (d e-c f)}-\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} (a (A d f (m+1)-B c f (m+n+2)+B d e (n+1))+b (A c f (n+1)-A d e (m+n+2)+B c e (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)} \]
Antiderivative was successfully verified.
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Rule 155
Rule 12
Rule 132
Rubi steps
\begin{align*} \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-3-m-n} \, dx &=\frac{(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac{\int (b (B c e (1+m)+A c f (1+n)-A d e (2+m+n))+a (A d f (1+m)+B d e (1+n)-B c f (2+m+n))) (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f) (d e-c f) (2+m+n)}\\ &=\frac{(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac{(b (B c e (1+m)+A c f (1+n)-A d e (2+m+n))+a (A d f (1+m)+B d e (1+n)-B c f (2+m+n))) \int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f) (d e-c f) (2+m+n)}\\ &=\frac{(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f) (d e-c f) (2+m+n)}-\frac{(b (B c e (1+m)+A c f (1+n)-A d e (2+m+n))+a (A d f (1+m)+B d e (1+n)-B c f (2+m+n))) (a+b x)^{1+m} (c+d x)^n \left (\frac{(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{-n} (e+f x)^{-1-m-n} \, _2F_1\left (1+m,-n;2+m;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f)^2 (d e-c f) (1+m) (2+m+n)}\\ \end{align*}
Mathematica [A] time = 0.31494, size = 223, normalized size = 0.85 \[ -\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-2} \left (\frac{(e+f x) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} (a (A d f (m+1)-B c f (m+n+2)+B d e (n+1))+b (A c f (n+1)-A d e (m+n+2)+B c e (m+1))) \, _2F_1\left (m+1,-n;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)}+(c+d x) (A f-B e)\right )}{(m+n+2) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.151, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( Bx+A \right ) \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-3-m-n}\, 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 (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}\,{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 (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}, 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 (B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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