Optimal. Leaf size=135 \[ \frac {(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (b (2 c f-d e (1-n))-a d f (n+1)) \, _2F_1\left (n,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f} \]
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Rubi [A] time = 0.08, 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 = {80, 70, 69} \[ \frac {(a+b x)^{-n} (e+f x)^{n+1} \left (-\frac {f (a+b x)}{b e-a f}\right )^n (-a d f (n+1)+2 b c f-b d e (1-n)) \, _2F_1\left (n,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{2 b f^2 (n+1)}+\frac {d (a+b x)^{1-n} (e+f x)^{n+1}}{2 b f} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
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 \[ \frac {(a+b x)^{-n} (e+f x)^{n+1} \left (\frac {\left (\frac {f (a+b x)}{a f-b e}\right )^n (-a d f (n+1)+2 b c f+b d e (n-1)) \, _2F_1\left (n,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{n+1}+d f (a+b x)\right )}{2 b f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \left (b x +a \right )^{-n} \left (f x +e \right )^{n}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n}}{{\left (b x + a\right )}^{n}}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^n\,\left (c+d\,x\right )}{{\left (a+b\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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