Optimal. Leaf size=154 \[ \frac {(a+b x)^{-n} (c+d x)^{n+1} \left (-\frac {d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{4 b d^2 (n+1)}-\frac {(b c-a d) (a+b x)^{1-n} (c+d x)^{n-1} \, _2F_1\left (2,1-n;2-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{4 b^3 d (1-n)} \]
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Rubi [C] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 0.73, 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} \[ \frac {(a+b x)^{2-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} F_1\left (2-n;-n-1,2;3-n;-\frac {d (a+b x)}{b c-a d},-\frac {2 d (a+b x)}{b c-a d}\right )}{b^2 (2-n) (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rubi steps
\begin {align*} \int \frac {(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^2} \, 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)^2} \, 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,2;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-n)}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 234, normalized size = 1.52 \[ -\frac {(a+b x)^{-n} (c+d x)^n \left (2 d (a+b x) (a d+b (c+2 d x)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;\frac {d (a+b x)}{a d-b c}\right )-(n-1) (b c-a d)^2 \left (\frac {d (a+b x)}{a d+b (c+2 d x)}\right )^n \left (\frac {b (c+d x)}{a d+b (c+2 d x)}\right )^{-n} F_1\left (1;-n,n;2;\frac {a d-b c}{a d+b (c+2 d x)},\frac {b c-a d}{b c+a d+2 b d x}\right )\right )}{8 b^2 d^2 (n-1) (a d+b (c+2 d x))} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{-n + 1} {\left (d x + c\right )}^{n + 1}}{4 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{-n + 1} {\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{-n +1} \left (d x +c \right )^{n +1}}{\left (2 b d x +a d +b c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{-n + 1} {\left (d x + c\right )}^{n + 1}}{{\left (2 \, b d x + b c + a d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{1-n}\,{\left (c+d\,x\right )}^{n+1}}{{\left (a\,d+b\,c+2\,b\,d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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