Optimal. Leaf size=245 \[ -\frac{\left (1-2 n^2\right ) (b c-a d)^2 (a+b x)^{-n} (c+d x)^n \left (-\frac{d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n;n+1;\frac{b (c+d x)}{b c-a d}\right )}{8 b^2 d^2 (1-n) n}+\frac{(b c-a d)^2 (a+b x)^{1-n} (c+d x)^{n-1} \, _2F_1\left (1,n-1;n;-\frac{b (c+d x)}{d (a+b x)}\right )}{8 b^3 d (1-n)}+\frac{(3-2 n) (b c-a d) (a+b x)^{2-n} (c+d x)^{n-1}}{8 b^3 (1-n)}+\frac{d (a+b x)^{3-n} (c+d x)^{n-1}}{4 b^3} \]
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Rubi [A] time = 0.270768, antiderivative size = 319, normalized size of antiderivative = 1.3, number of steps used = 10, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {105, 70, 69, 131} \[ -\frac{(b c-a d)^2 (a+b x)^{-n} (c+d x)^n \, _2F_1\left (1,-n;1-n;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^2 d^2 n}+\frac{(b c-a d)^2 (a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{8 b^2 d^2 n}-\frac{(b c-a d) (a+b x)^{-n} (c+d x)^{n+1} \left (-\frac{d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{4 b d^2 (n+1)}+\frac{(a+b x)^{-n} (c+d x)^{n+2} \left (-\frac{d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n,n+2;n+3;\frac{b (c+d x)}{b c-a d}\right )}{2 d^2 (n+2)} \]
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)^{1-n} (c+d x)^{1+n}}{b c+a d+2 b d x} \, dx &=\frac{\int (a+b x)^{-n} (c+d x)^{1+n} \, dx}{2 d}-\frac{(b c-a d) \int \frac{(a+b x)^{-n} (c+d x)^{1+n}}{b c+a d+2 b d x} \, dx}{2 d}\\ &=-\frac{(b c-a d) \int (a+b x)^{-n} (c+d x)^n \, dx}{4 b d}-\frac{(b c-a d)^2 \int \frac{(a+b x)^{-n} (c+d x)^n}{b c+a d+2 b d x} \, dx}{4 b d}+\frac{\left ((a+b x)^{-n} \left (\frac{d (a+b x)}{-b c+a d}\right )^n\right ) \int (c+d x)^{1+n} \left (-\frac{a d}{b c-a d}-\frac{b d x}{b c-a d}\right )^{-n} \, dx}{2 d}\\ &=\frac{(a+b x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^n (c+d x)^{2+n} \, _2F_1\left (n,2+n;3+n;\frac{b (c+d x)}{b c-a d}\right )}{2 d^2 (2+n)}-\frac{(b c-a d)^2 \int (a+b x)^{-1-n} (c+d x)^n \, dx}{8 b d^2}+\frac{(b c-a d)^3 \int \frac{(a+b x)^{-1-n} (c+d x)^n}{b c+a d+2 b d x} \, dx}{8 b d^2}-\frac{\left ((b c-a d) (a+b x)^{-n} \left (\frac{d (a+b x)}{-b c+a d}\right )^n\right ) \int (c+d x)^n \left (-\frac{a d}{b c-a d}-\frac{b d x}{b c-a d}\right )^{-n} \, dx}{4 b d}\\ &=-\frac{(b c-a d)^2 (a+b x)^{-n} (c+d x)^n \, _2F_1\left (1,-n;1-n;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^2 d^2 n}-\frac{(b c-a d) (a+b x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^n (c+d x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac{b (c+d x)}{b c-a d}\right )}{4 b d^2 (1+n)}+\frac{(a+b x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^n (c+d x)^{2+n} \, _2F_1\left (n,2+n;3+n;\frac{b (c+d x)}{b c-a d}\right )}{2 d^2 (2+n)}-\frac{\left ((b c-a d)^2 (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{-1-n} \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \, dx}{8 b d^2}\\ &=-\frac{(b c-a d)^2 (a+b x)^{-n} (c+d x)^n \, _2F_1\left (1,-n;1-n;-\frac{d (a+b x)}{b (c+d x)}\right )}{8 b^2 d^2 n}+\frac{(b c-a d)^2 (a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{8 b^2 d^2 n}-\frac{(b c-a d) (a+b x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^n (c+d x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac{b (c+d x)}{b c-a d}\right )}{4 b d^2 (1+n)}+\frac{(a+b x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^n (c+d x)^{2+n} \, _2F_1\left (n,2+n;3+n;\frac{b (c+d x)}{b c-a d}\right )}{2 d^2 (2+n)}\\ \end{align*}
Mathematica [A] time = 0.645188, size = 257, normalized size = 1.05 \[ \frac{(a d-b c) (a+b x)^{-n} (c+d x)^n \left (\frac{\left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (4 d n (n+1) (a+b x) \, _2F_1\left (-n-1,1-n;2-n;\frac{d (a+b x)}{a d-b c}\right )-(n-1) \left ((n+1) (b c-a d) \, _2F_1\left (-n,-n;1-n;\frac{d (a+b x)}{a d-b c}\right )-2 b n (c+d x) \left (-\frac{b d (a+b x) (c+d x)}{(b c-a d)^2}\right )^n \, _2F_1\left (n,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )\right )\right )}{n^2-1}+(b c-a d) \, _2F_1\left (1,-n;1-n;-\frac{d (a+b x)}{b (c+d x)}\right )\right )}{8 b^2 d^2 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{1+n} \left ( bx+a \right ) ^{1-n}}{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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{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 )}^{-n + 1}{\left (d x + c\right )}^{n + 1}}{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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