Optimal. Leaf size=277 \[ \frac {B (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{f (m+1) (b e-a f)}-\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{f (m+1) (b e-a f)} \]
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Rubi [A] time = 0.15, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {159, 140, 139, 138, 132} \[ \frac {B (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{f (m+1) (b e-a f)}-\frac {(a+b x)^{m+1} (B e-A f) (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{f (m+1) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 132
Rule 138
Rule 139
Rule 140
Rule 159
Rubi steps
\begin {align*} \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-2-m-n} \, dx &=\frac {B \int (a+b x)^m (c+d x)^n (e+f x)^{-1-m-n} \, dx}{f}+\frac {(-B e+A f) \int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx}{f}\\ &=-\frac {(B e-A f) (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 )}{f (b e-a f) (1+m)}+\frac {\left (B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^{-1-m-n} \, dx}{f}\\ &=-\frac {(B e-A f) (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 )}{f (b e-a f) (1+m)}+\frac {\left (b B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^{-1-m-n} \, dx}{f (b e-a f)}\\ &=\frac {B (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (1+m;-n,1+m+n;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{f (b e-a f) (1+m)}-\frac {(B e-A f) (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 )}{f (b e-a f) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 215, normalized size = 0.78 \[ -\frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^n \left ((A f-B e) \, _2F_1\left (m+1,-n;m+2;\frac {(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )+B (e+f x) \left (\frac {b (e+f x)}{b e-a f}\right )^m F_1\left (m+1;-n,m+n+1;m+2;\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )\right )}{f (m+1) (a f-b e)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 2}, x\right ) \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 2}\,{d x} \]
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 \[ \int \left (B x +A \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{-m -n -2}\, dx \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 2}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^{m+n+2}} \,d x \]
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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