Optimal. Leaf size=558 \[ \frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} ((m+n+2) (b d e ((m+n+3) (A (-a d f-b c f+b d e)+a B c f)-(B e-A f) (a d (n+1)+b c (m+1)))-f (a d+b c) ((m+n+3) (A (-a d f-b c f+b d e)+a B c f)-(B e-A f) (a d (n+1)+b c (m+1)))+a b c d f (B e-A f))-(a d (n+1)+b c (m+1)) (a f (A d f (m+2)+B (d e (n+1)-c f (m+n+3)))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))) \, _2F_1\left (m+1,-n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (m+n+3) (b e-a f)^3 (d e-c f)^2}+\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^{n+1} (e+f x)^{-m-n-3}}{(m+n+3) (b e-a f) (d e-c f)}+\frac {(a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2} (a f (A d f (m+2)+B (d e (n+1)-c f (m+n+3)))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))}{(m+n+2) (m+n+3) (b e-a f)^2 (d e-c f)^2} \]
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Rubi [A] time = 0.98, antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {155, 12, 132} \[ \frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} ((m+n+2) (-b d e (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+f (a d+b c) (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+a b c d f (B e-A f))-(a d (n+1)+b c (m+1)) (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))) \, _2F_1\left (m+1,-n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (m+n+3) (b e-a f)^3 (d e-c f)^2}+\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^{n+1} (e+f x)^{-m-n-3}}{(m+n+3) (b e-a f) (d e-c f)}+\frac {(a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2} (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))}{(m+n+2) (m+n+3) (b e-a f)^2 (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 132
Rule 155
Rubi steps
\begin {align*} \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-4-m-n} \, dx &=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}-\frac {\int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))-b d (B e-A f) x) \, dx}{(b e-a f) (d e-c f) (3+m+n)}\\ &=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {\int ((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}\\ &=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n))))) \int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}\\ &=\frac {(B e-A f) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-3-m-n}}{(b e-a f) (d e-c f) (3+m+n)}+\frac {(a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+m+n)))) (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)^{-2-m-n}}{(b e-a f)^2 (d e-c f)^2 (2+m+n) (3+m+n)}+\frac {((2+m+n) (a b c d f (B e-A f)-b d e (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n)))+(b c+a d) f (b (B c e (1+m)+A c f (2+n)-A d e (3+m+n))+a (A d f (2+m)+B d e (1+n)-B c f (3+m+n))))-(b c (1+m)+a d (1+n)) (a f (A d f (2+m)+B d e (1+n)-B c f (3+m+n))+b (B e (d e+c f (1+m))+A f (c f (2+n)-d e (4+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)^3 (d e-c f)^2 (1+m) (2+m+n) (3+m+n)}\\ \end {align*}
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Mathematica [A] time = 1.94, size = 508, normalized size = 0.91 \[ -\frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-3} \left (-\frac {(e+f x)^2 \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} ((m+n+2) (-b d e (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+f (a d+b c) (a (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A c f (n+2)-A d e (m+n+3)+B c e (m+1)))+a b c d f (B e-A f))-(a d (n+1)+b c (m+1)) (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))) \, _2F_1\left (m+1,-n;m+2;\frac {(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)}-\frac {(c+d x) (e+f x) (a f (A d f (m+2)-B c f (m+n+3)+B d e (n+1))+b (A f (c f (n+2)-d e (m+n+4))+B e (c f (m+1)+d e)))}{(m+n+2) (b e-a f) (d e-c f)}-(c+d x) (B e-A f)\right )}{(m+n+3) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 4}, x\right ) \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{-m -n -4}\, dx \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 4}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^{m+n+4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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