Optimal. Leaf size=135 \[ \frac {d (a+b x)^{1-n} (e+f x)^{1+n}}{2 b f}+\frac {(b (2 c f-d e (1-n))-a d f (1+n)) (a+b x)^{-n} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (e+f x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (1+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {81, 72, 71}
\begin {gather*} \frac {(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (-a d f (n+1)+2 b c f-b d e (1-n)) \, _2F_1\left (n,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 81
Rubi steps
\begin {align*} \int (a+b x)^{-n} (c+d x) (e+f x)^n \, dx &=\frac {d (a+b x)^{1-n} (e+f x)^{1+n}}{2 b f}+\frac {(2 b c f-d (b e (1-n)+a f (1+n))) \int (a+b x)^{-n} (e+f x)^n \, dx}{2 b f}\\ &=\frac {d (a+b x)^{1-n} (e+f x)^{1+n}}{2 b f}+\frac {\left ((2 b c f-d (b e (1-n)+a f (1+n))) (a+b x)^{-n} \left (\frac {f (a+b x)}{-b e+a f}\right )^n\right ) \int (e+f x)^n \left (-\frac {a f}{b e-a f}-\frac {b f x}{b e-a f}\right )^{-n} \, dx}{2 b f}\\ &=\frac {d (a+b x)^{1-n} (e+f x)^{1+n}}{2 b f}+\frac {(2 b c f-b d e (1-n)-a d f (1+n)) (a+b x)^{-n} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (e+f x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 108, normalized size = 0.80 \begin {gather*} \frac {(a+b x)^{-n} (e+f x)^{1+n} \left (d f (a+b x)+\frac {(2 b c f+b d e (-1+n)-a d f (1+n)) \left (\frac {f (a+b x)}{-b e+a f}\right )^n \, _2F_1\left (n,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{1+n}\right )}{2 b f^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (d x +c \right ) \left (f x +e \right )^{n} \left (b x +a \right )^{-n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^n\,\left (c+d\,x\right )}{{\left (a+b\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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