Optimal. Leaf size=71 \[ \frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (2-n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {133}
\begin {gather*} \frac {(a+b x)^{2-n} (c+d x)^{n-2} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (2-n) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
Rubi steps
\begin {align*} \int \frac {(a+b x)^{1-n} (c+d x)^{1+n}}{(b c+a d+2 b d x)^4} \, dx &=\frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 70, normalized size = 0.99 \begin {gather*} -\frac {(a+b x)^{2-n} (c+d x)^{-2+n} \, _2F_1\left (4,2-n;3-n;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (-2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1-n} \left (d x +c \right )^{1+n}}{\left (2 b d x +a d +b c \right )^{4}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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