Optimal. Leaf size=126 \[ -\frac{e^2 (d+e x)^{m+1} (b (3 B d-A e (2-m))-a B e (m+1)) \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{3 b (m+1) (b d-a e)^4}-\frac{(A b-a B) (d+e x)^{m+1}}{3 b (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0725696, antiderivative size = 125, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {27, 78, 68} \[ -\frac{e^2 (d+e x)^{m+1} (-a B e (m+1)-A b e (2-m)+3 b B d) \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{3 b (m+1) (b d-a e)^4}-\frac{(A b-a B) (d+e x)^{m+1}}{3 b (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 68
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(A+B x) (d+e x)^m}{(a+b x)^4} \, dx\\ &=-\frac{(A b-a B) (d+e x)^{1+m}}{3 b (b d-a e) (a+b x)^3}+\frac{(3 b B d-A b e (2-m)-a B e (1+m)) \int \frac{(d+e x)^m}{(a+b x)^3} \, dx}{3 b (b d-a e)}\\ &=-\frac{(A b-a B) (d+e x)^{1+m}}{3 b (b d-a e) (a+b x)^3}-\frac{e^2 (3 b B d-A b e (2-m)-a B e (1+m)) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac{b (d+e x)}{b d-a e}\right )}{3 b (b d-a e)^4 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0807949, size = 109, normalized size = 0.87 \[ \frac{(d+e x)^{m+1} \left (\frac{a B-A b}{(a+b x)^3}-\frac{e^2 (-a B e (m+1)+A b e (m-2)+3 b B d) \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3}\right )}{3 b (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{ \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 )}{\left (e x + d\right )}^{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 )}{\left (e x + d\right )}^{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{\left (A + B x\right ) \left (d + e x\right )^{m}}{\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 )}{\left (e x + d\right )}^{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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