Optimal. Leaf size=37 \[ -\frac {(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} -\frac {(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int (a+b x)^{-2-n} (c+d x)^n \, dx &=-\frac {(a+b x)^{-1-n} (c+d x)^{1+n}}{(b c-a d) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.03 \begin {gather*} \frac {(a+b x)^{-n-1} (c+d x)^{n+1}}{(-n-1) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^{-2-n} (c+d x)^n \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.33, size = 59, normalized size = 1.59 \begin {gather*} -\frac {{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (b x + a\right )}^{-n - 2} {\left (d x + c\right )}^{n}}{b c - a d + {\left (b c - a d\right )} n} \end {gather*}
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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 2} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 1.11 \begin {gather*} \frac {\left (b x +a \right )^{-n -1} \left (d x +c \right )^{n +1}}{a d n -b c n +a d -b c} \end {gather*}
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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 2} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 97, normalized size = 2.62 \begin {gather*} \frac {\frac {a\,c\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {x\,\left (a\,d+b\,c\right )\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}}{{\left (a+b\,x\right )}^{n+2}} \end {gather*}
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 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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