Optimal. Leaf size=81 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]
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Rubi [A] time = 0.0318195, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 78, 64} \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 64
Rubi steps
\begin{align*} \int \frac{x^m (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x^m (A+B x)}{(a+b x)^4} \, dx\\ &=\frac{(A b-a B) x^{1+m}}{3 a b (a+b x)^3}-\frac{(A b (-2+m)-a B (1+m)) \int \frac{x^m}{(a+b x)^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^{1+m}}{3 a b (a+b x)^3}+\frac{(A b (2-m)+a B (1+m)) x^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac{b x}{a}\right )}{3 a^4 b (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0347746, size = 71, normalized size = 0.88 \[ \frac{x^{m+1} \left (\frac{a^3 (A b-a B)}{(a+b x)^3}-\frac{(A b (m-2)-a B (m+1)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{m+1}\right )}{3 a^4 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( Bx+A \right ) }{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \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 )} x^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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 )} x^{m}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, 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{x^{m} \left (A + B x\right )}{\left (a + b x\right )^{4}}\, 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 )} x^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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