Optimal. Leaf size=118 \[ \frac {d (a+b x)^{-n} (e+f x)^n \left (-\frac {f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac {b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
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Rubi [A] time = 0.05, 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 = {79, 70, 69} \[ \frac {d (a+b x)^{-n} (e+f x)^n \left (-\frac {f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac {b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 79
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 = 0.20, size = 105, normalized size = 0.89 \[ \frac {(a+b x)^{-n} (e+f x)^n \left (\frac {f (a+b x) (d e-c f)}{(n-1) (e+f x) (b e-a f)}+\frac {d \left (\frac {f (a+b x)}{a f-b e}\right )^n \, _2F_1\left (n,n;n+1;\frac {b (e+f x)}{b e-a f}\right )}{n}\right )}{f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \left (b x +a \right )^{-n} \left (f x +e \right )^{n -2}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^{n-2}\,\left (c+d\,x\right )}{{\left (a+b\,x\right )}^n} \,d x \]
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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