Optimal. Leaf size=72 \[ \frac{(a+b x)^{n+3} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+3;n+4;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+3)} \]
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Rubi [A] time = 0.0227955, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {70, 69} \[ \frac{(a+b x)^{n+3} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+3;n+4;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+3)} \]
Antiderivative was successfully verified.
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Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{2+n} (c+d x)^{-n} \, dx &=\left ((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 )^{-n} \, dx\\ &=\frac{(a+b x)^{3+n} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,3+n;4+n;-\frac{d (a+b x)}{b c-a d}\right )}{b (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0373166, size = 92, normalized size = 1.28 \[ -\frac{(b c-a d)^2 (a+b x)^n (c+d x)^{1-n} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n-2,1-n;2-n;\frac{b (c+d x)}{b c-a d}\right )}{d^3 (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{2+n}}{ \left ( dx+c \right ) ^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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