Optimal. Leaf size=72 \[ \frac {(a+b x)^{3+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,3+n;4+n;-\frac {d (a+b x)}{b c-a d}\right )}{b (3+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71}
\begin {gather*} \frac {(a+b x)^{n+3} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+3;n+4;-\frac {d (a+b x)}{b c-a d}\right )}{b (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rubi steps
\begin {align*} \int (a+b x)^{2+n} (c+d x)^{-n} \, dx &=\left ((c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^{2+n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx\\ &=\frac {(a+b x)^{3+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,3+n;4+n;-\frac {d (a+b x)}{b c-a d}\right )}{b (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 92, normalized size = 1.28 \begin {gather*} -\frac {(b c-a d)^2 (a+b x)^n \left (\frac {d (a+b x)}{-b c+a d}\right )^{-n} (c+d x)^{1-n} \, _2F_1\left (-2-n,1-n;2-n;\frac {b (c+d x)}{b c-a d}\right )}{d^3 (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{2+n} \left (d x +c \right )^{-n}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 )}^{n+2}}{{\left (c+d\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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