Optimal. Leaf size=52 \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]
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Rubi [A] time = 0.0094082, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {68} \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^n}{(c+d x)^2} \, dx &=\frac{b (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;-\frac{d (a+b x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0130413, size = 52, normalized size = 1. \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{2}}}\, 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}}{{\left (d x + c\right )}^{2}}\,{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}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, dx \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}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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