Optimal. Leaf size=45 \[ \frac {(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]
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Rubi [A] time = 0.02, 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 = {186, 69} \[ \frac {(c+d x)^{n+1} \, _2F_1\left (-m,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 186
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 = 0.04, size = 88, normalized size = 1.96 \[ \frac {(a+b x) (c+d x)^n \left (\frac {d (a+b x)}{a d-b c}\right )^m \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;\frac {d (a+b x)}{a d-b c}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )}^{n} \left (-\frac {b d x + a d}{b c - a d}\right )^{m}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{n} \left (-\frac {{\left (b x + a\right )} d}{b c - a d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (\frac {\left (b x +a \right ) d}{a d -b c}\right )^{m} \left (d x +c \right )^{n}\, dx \]
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 \[ \int {\left (d x + c\right )}^{n} \left (-\frac {{\left (b x + a\right )} d}{b c - a d}\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (c+d\,x\right )}^n\,{\left (\frac {d\,\left (a+b\,x\right )}{a\,d-b\,c}\right )}^m \,d x \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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