Optimal. Leaf size=134 \[ \frac {(c+d x)^{n+1} (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n (2 a d f-b (c f (1-n)+d e (n+1))) \, _2F_1\left (n,n+1;n+2;-\frac {f (c+d x)}{d e-c f}\right )}{2 d^2 f (n+1)}+\frac {b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f} \]
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Rubi [A] time = 0.08, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 = {80, 70, 69} \[ \frac {(c+d x)^{n+1} (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n (2 a d f-b c f (1-n)-b d e (n+1)) \, _2F_1\left (n,n+1;n+2;-\frac {f (c+d x)}{d e-c f}\right )}{2 d^2 f (n+1)}+\frac {b (c+d x)^{n+1} (e+f x)^{1-n}}{2 d f} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^n (e+f x)^{-n} \, dx &=\frac {b (c+d x)^{1+n} (e+f x)^{1-n}}{2 d f}+\frac {(2 a d f-b (c f (1-n)+d e (1+n))) \int (c+d x)^n (e+f x)^{-n} \, dx}{2 d f}\\ &=\frac {b (c+d x)^{1+n} (e+f x)^{1-n}}{2 d f}+\frac {\left ((2 a d f-b (c f (1-n)+d e (1+n))) (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n\right ) \int (c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^{-n} \, dx}{2 d f}\\ &=\frac {b (c+d x)^{1+n} (e+f x)^{1-n}}{2 d f}+\frac {(2 a d f-b c f (1-n)-b d e (1+n)) (c+d x)^{1+n} (e+f x)^{-n} \left (\frac {d (e+f x)}{d e-c f}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {f (c+d x)}{d e-c f}\right )}{2 d^2 f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 109, normalized size = 0.81 \[ \frac {(c+d x)^{n+1} (e+f x)^{-n} \left (b d (e+f x)-\frac {\left (\frac {d (e+f x)}{d e-c f}\right )^n (-2 a d f-b c f (n-1)+b d e (n+1)) \, _2F_1\left (n,n+1;n+2;\frac {f (c+d x)}{c f-d e}\right )}{n+1}\right )}{2 d^2 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right ) \left (d x +c \right )^{n} \left (f x +e \right )^{-n}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{n}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,x\right )\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^n} \,d x \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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