Optimal. Leaf size=52 \[ \frac{b (a+b x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]
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Rubi [A] time = 0.0110129, 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} \[ \frac{b (a+b x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{(e+f x)^2} \, dx &=\frac{b (a+b x)^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac{f (a+b x)}{b e-a f}\right )}{(b e-a f)^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0170137, size = 52, normalized size = 1. \[ \frac{b (a+b x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( fx+e \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 )}^{m}}{{\left (f x + e\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 )}^{m}}{f^{2} x^{2} + 2 \, e f x + e^{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 )^{m}}{\left (e + f 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 )}^{m}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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