Optimal. Leaf size=118 \[ -\frac {(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac {d (a+b x)^{-n} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (e+f x)^n \, _2F_1\left (n,n;1+n;\frac {b (e+f x)}{b e-a f}\right )}{f^2 n} \]
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Rubi [A]
time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 72, 71}
\begin {gather*} \frac {d (a+b x)^{-n} (e+f x)^n \left (-\frac {f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac {b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 80
Rubi steps
\begin {align*} \int (a+b x)^{-n} (c+d x) (e+f x)^{-2+n} \, dx &=-\frac {(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac {d \int (a+b x)^{-n} (e+f x)^{-1+n} \, dx}{f}\\ &=-\frac {(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac {\left (d (a+b x)^{-n} \left (\frac {f (a+b x)}{-b e+a f}\right )^n\right ) \int (e+f x)^{-1+n} \left (-\frac {a f}{b e-a f}-\frac {b f x}{b e-a f}\right )^{-n} \, dx}{f}\\ &=-\frac {(d e-c f) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f) (1-n)}+\frac {d (a+b x)^{-n} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (e+f x)^n \, _2F_1\left (n,n;1+n;\frac {b (e+f x)}{b e-a f}\right )}{f^2 n}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 125, normalized size = 1.06 \begin {gather*} -\frac {(a+b x)^{-n} (e+f x)^{-1+n} \left (-d e f n (a+b x)+c f^2 n (a+b x)-d (b e-a f) (-1+n) \left (\frac {f (a+b x)}{-b e+a f}\right )^n (e+f x) \, _2F_1\left (n,n;1+n;\frac {b (e+f x)}{b e-a f}\right )\right )}{f^2 (b e-a f) (-1+n) n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d x +c \right ) \left (f x +e \right )^{-2+n} \left (b x +a \right )^{-n}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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-2}\,\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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