Optimal. Leaf size=230 \[ -\frac{\left (1-2 n^2\right ) (a+b x)^{-n} (c+d x)^n \, _2F_1\left (1,n;n+1;-\frac{b (c+d x)}{d (a+b x)}\right )}{8 b^2 d^2 n}+\frac{(a+b x)^{-n} (c+d x)^n \left (-\frac{d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (c+d x)}{b c-a d}\right )}{8 b^2 d^2 n}-\frac{(2 n+1) (a+b x)^{1-n} (c+d x)^n}{8 b^2 d (a d+b c+2 b d x)}-\frac{(b c-a d) (a+b x)^{1-n} (c+d x)^n}{8 b^2 d (a d+b c+2 b d x)^2} \]
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Rubi [C] time = 0.071605, antiderivative size = 113, normalized size of antiderivative = 0.49, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {137, 136} \[ \frac{(a+b x)^{2-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} F_1\left (2-n;-n-1,3;3-n;-\frac{d (a+b x)}{b c-a d},-\frac{2 d (a+b x)}{b c-a d}\right )}{b^2 (2-n) (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^3} \, dx &=\frac{\left ((b c-a d) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int \frac{(a+b x)^{1-n} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{1+n}}{(b c+a d+2 b d x)^3} \, dx}{b}\\ &=\frac{(a+b x)^{2-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} F_1\left (2-n;-1-n,3;3-n;-\frac{d (a+b x)}{b c-a d},-\frac{2 d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d)^2 (2-n)}\\ \end{align*}
Mathematica [C] time = 0.67597, size = 239, normalized size = 1.04 \[ \frac{(a+b x)^{-n} (c+d x)^n \left (\frac{(b c-a d)^3 \left (\frac{d (a+b x)}{a d+b (c+2 d x)}\right )^n \left (\frac{b (c+d x)}{a d+b (c+2 d x)}\right )^{-n} F_1\left (2;-n,n;3;\frac{a d-b c}{a d+b (c+2 d x)},\frac{b c-a d}{b c+a d+2 b d x}\right )}{(a d+b (c+2 d x))^2}-\frac{4 b (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^n F_1\left (n+1;n,1;n+2;\frac{b (c+d x)}{b c-a d},\frac{2 b (c+d x)}{b c-a d}\right )}{n+1}\right )}{16 b^2 d^2 (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{1+n} \left ( bx+a \right ) ^{1-n}}{ \left ( 2\,bdx+ad+bc \right ) ^{3}}}\, 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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{3}}\,{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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{8 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 12 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}, 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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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