Optimal. Leaf size=231 \[ -\frac{\left (2 m^2-4 m+1\right ) (b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{8 b^3 m (m+1)}+\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;-\frac{b (c+d x)}{d (a+b x)}\right )}{8 b^3 d m}+\frac{(2 m+1) (b c-a d) (a+b x)^{m+1} (c+d x)^{-m}}{8 b^3 m}+\frac{d (a+b x)^{m+2} (c+d x)^{-m}}{4 b^3} \]
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Rubi [A] time = 0.268593, antiderivative size = 314, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {105, 70, 69, 131} \[ -\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^3 d m}+\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^3 (m+1)}+\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{8 b^3 d m}+\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^3 (m+1)} \]
Antiderivative was successfully verified.
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Rule 105
Rule 70
Rule 69
Rule 131
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{2-m}}{b c+a d+2 b d x} \, dx &=\frac{\int (a+b x)^m (c+d x)^{1-m} \, dx}{2 b}+\frac{(b c-a d) \int \frac{(a+b x)^m (c+d x)^{1-m}}{b c+a d+2 b d x} \, dx}{2 b}\\ &=\frac{(b c-a d) \int (a+b x)^m (c+d x)^{-m} \, dx}{4 b^2}+\frac{(b c-a d)^2 \int \frac{(a+b x)^m (c+d x)^{-m}}{b c+a d+2 b d x} \, dx}{4 b^2}+\frac{\left ((b c-a d) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{1-m} \, dx}{2 b^2}\\ &=\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-1+m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^3 (1+m)}+\frac{(b c-a d)^2 \int (a+b x)^{-1+m} (c+d x)^{-m} \, dx}{8 b^2 d}-\frac{(b c-a d)^3 \int \frac{(a+b x)^{-1+m} (c+d x)^{-m}}{b c+a d+2 b d x} \, dx}{8 b^2 d}+\frac{\left ((b c-a d) (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m} \, dx}{4 b^2}\\ &=-\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^3 d m}+\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-1+m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^3 (1+m)}+\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^3 (1+m)}+\frac{\left ((b c-a d)^2 (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^{-1+m} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-m} \, dx}{8 b^2 d}\\ &=-\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;1+m;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^3 d m}+\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-1+m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^3 (1+m)}+\frac{(b c-a d)^2 (a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;1+m;-\frac{d (a+b x)}{b c-a d}\right )}{8 b^3 d m}+\frac{(b c-a d) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{4 b^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.28238, size = 243, normalized size = 1.05 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (-4 d m (a+b x) (a d-b c) \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )-(b c-a d) \left ((m+1) (b c-a d) \left (\, _2F_1\left (1,m;m+1;-\frac{d (a+b x)}{b (c+d x)}\right )-\left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;\frac{d (a+b x)}{a d-b c}\right )\right )-2 d m (a+b x) \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;\frac{d (a+b x)}{a d-b c}\right )\right )\right )}{8 b^3 d m (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{2\,bdx+ad+bc}}\, 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 (d x + c\right )}^{-m + 2}}{2 \, b d x + b c + a d}\,{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}{\left (d x + c\right )}^{-m + 2}}{2 \, b d x + b c + a d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (d x + c\right )}^{-m + 2}}{2 \, b d x + b c + a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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