Optimal. Leaf size=108 \[ \frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;-\frac {d (a+b x)}{b c-a d}\right )}{n}-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;n+1;\frac {c (a+b x)}{a (c+d x)}\right )}{n} \]
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Rubi [A] time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {105, 70, 69, 131} \[ \frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;-\frac {d (a+b x)}{b c-a d}\right )}{n}-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;n+1;\frac {c (a+b x)}{a (c+d x)}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 105
Rule 131
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^{-n}}{x} \, dx &=a \int \frac {(a+b x)^{-1+n} (c+d x)^{-n}}{x} \, dx+b \int (a+b x)^{-1+n} (c+d x)^{-n} \, dx\\ &=-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )}{n}+\left (b (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{-1+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=-\frac {(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;1+n;\frac {c (a+b x)}{a (c+d x)}\right )}{n}+\frac {(a+b x)^n (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;1+n;-\frac {d (a+b x)}{b c-a d}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 89, normalized size = 0.82 \[ \frac {(a+b x)^n (c+d x)^{-n} \left (\left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;\frac {d (a+b x)}{a d-b c}\right )-\, _2F_1\left (1,n;n+1;\frac {c (a+b x)}{a (c+d x)}\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{-n}}{x}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^n}{x\,{\left (c+d\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{n} \left (c + d x\right )^{- n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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