Optimal. Leaf size=83 \[ -\frac {(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n-1;n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {70, 69} \[ -\frac {(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n-1;n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rubi steps
\begin {align*} \int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx &=\frac {\left ((b c-a d) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{-2+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{1-n} \, dx}{b}\\ &=-\frac {(b c-a d) (a+b x)^{-1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (-1+n,-1+n;n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.90 \[ \frac {(a+b x)^{n-1} (c+d x)^{1-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{n-1} \, _2F_1\left (n-1,n-1;n;\frac {d (a+b x)}{a d-b c}\right )}{b (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{n -2} \left (d x +c \right )^{-n +1}\, dx \]
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 \[ \int {\left (b x + a\right )}^{n - 2} {\left (d x + c\right )}^{-n + 1}\,{d x} \]
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 \[ \int {\left (a+b\,x\right )}^{n-2}\,{\left (c+d\,x\right )}^{1-n} \,d x \]
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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