Optimal. Leaf size=51 \[ \frac{e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2} \]
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Rubi [A] time = 0.0176339, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 68} \[ \frac{e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^m}{(a+b x)^2} \, dx\\ &=\frac{e (d+e x)^{1+m} \, _2F_1\left (2,1+m;2+m;\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0139702, size = 52, normalized size = 1.02 \[ \frac{e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{b (d+e x)}{a e-b d}\right )}{(m+1) (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.005, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{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 (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{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 (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{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 (d + e x\right )^{m}}{\left (a + b 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 (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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