Optimal. Leaf size=36 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 36, 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} \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int (a+b x)^n (c+d x)^{-2-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 = 0.01, size = 36, normalized size = 1.00 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 58, normalized size = 1.61 \[ \frac {{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}}{b c - a d + {\left (b c - a d\right )} n} \]
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 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 1.17 \[ -\frac {\left (b x +a \right )^{n +1} \left (d x +c \right )^{-n -1}}{a d n -b c n +a d -b c} \]
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 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 98, normalized size = 2.72 \[ -\frac {\frac {a\,c\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {x\,\left (a\,d+b\,c\right )\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}}{{\left (c+d\,x\right )}^{n+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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