Optimal. Leaf size=85 \[ \frac{B (d+e x)^{m+1}}{b e (m+1)}-\frac{(A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \]
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Rubi [A] time = 0.0429038, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {80, 68} \[ \frac{B (d+e x)^{m+1}}{b e (m+1)}-\frac{(A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^m}{a+b x} \, dx &=\frac{B (d+e x)^{1+m}}{b e (1+m)}+\frac{(A b e (1+m)-a B e (1+m)) \int \frac{(d+e x)^m}{a+b x} \, dx}{b e (1+m)}\\ &=\frac{B (d+e x)^{1+m}}{b e (1+m)}-\frac{(A b-a B) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b (d+e x)}{b d-a e}\right )}{b (b d-a e) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0285859, size = 78, normalized size = 0.92 \[ \frac{(d+e x)^{m+1} \left ((a B e-A b e) \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )+B (b d-a e)\right )}{b e (m+1) (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{bx+a}}\, 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}}{b x + a}\,{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 x + a}, 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}}{a + b x}\, 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}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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