Optimal. Leaf size=39 \[ -\frac {(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 39, 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)^{1-n}}{(1-n) (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 = 36, normalized size = 0.92 \begin {gather*} \frac {(a+b x)^{n-1} (c+d x)^{1-n}}{(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 = 1.32, size = 60, normalized size = 1.54 \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 (b c - a d - {\left (b c - a d\right )} n\right )} {\left (d x + c\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 \frac {{\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 = 45, normalized size = 1.15 \begin {gather*} -\frac {\left (d x +c \right ) \left (b x +a \right )^{n -1} \left (d x +c \right )^{-n}}{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 \frac {{\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 = 0.56, size = 102, normalized size = 2.62 \begin {gather*} -{\left (a+b\,x\right )}^{n-2}\,\left (\frac {a\,c}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {x\,\left (a\,d+b\,c\right )}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {b\,d\,x^2}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}\right ) \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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