Optimal. Leaf size=137 \[ \frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]
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Rubi [A] time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {140, 139, 138} \[ \frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (m+1;-n,m+n+1;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^n (e+f x)^{-1-m-n} \, dx &=\left ((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\\ &=\frac {\left (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}{b e-a f}\\ &=\frac {(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 )}{(b e-a f) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 130, normalized size = 0.95 \[ \frac {(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n+1} F_1\left (m+1;-n,m+n+1;m+2;\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 1}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{-m -n -1}\, dx \]
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 \[ \int {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^{m+n+1}} \,d x \]
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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