Optimal. Leaf size=45 \[ \frac {(c+d x)^{1+n} \, _2F_1\left (-m,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {192, 71}
\begin {gather*} \frac {(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 192
Rubi steps
\begin {align*} \int \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \, dx &=\int (c+d x)^n \left (-\frac {a d}{b c-a d}-\frac {b d x}{b c-a d}\right )^m \, dx\\ &=\frac {(c+d x)^{1+n} \, _2F_1\left (-m,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 88, normalized size = 1.96 \begin {gather*} \frac {(a+b x) \left (\frac {d (a+b x)}{-b c+a d}\right )^m (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (\frac {d \left (b x +a \right )}{a d -b c}\right )^{m} \left (d x +c \right )^{n}\, dx\]
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 = -0.02 \begin {gather*} \int {\left (c+d\,x\right )}^n\,{\left (\frac {d\,\left (a+b\,x\right )}{a\,d-b\,c}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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