Optimal. Leaf size=141 \[ \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac {B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]
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Rubi [A] time = 0.09, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {80, 70, 69} \[ \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (A b d (m+n+2)-B (a d (n+1)+b c (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (m+1) (m+n+2)}+\frac {B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
Rubi steps
\begin {align*} \int (a+b x)^m (A+B x) (c+d x)^n \, dx &=\frac {B (a+b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (A-\frac {B (b c (1+m)+a d (1+n))}{b d (2+m+n)}\right ) \int (a+b x)^m (c+d x)^n \, dx\\ &=\frac {B (a+b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\left (\left (A-\frac {B (b c (1+m)+a d (1+n))}{b d (2+m+n)}\right ) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \, dx\\ &=\frac {B (a+b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\frac {\left (A-\frac {B (b c (1+m)+a d (1+n))}{b d (2+m+n)}\right ) (a+b x)^{1+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 {align*}
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Mathematica [A] time = 0.11, size = 117, normalized size = 0.83 \[ \frac {(a+b x)^{m+1} (c+d x)^n \left (b B (c+d x)-\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (a B d (n+1)-A b d (m+n+2)+b B c (m+1)) \, _2F_1\left (m+1,-n;m+2;\frac {d (a+b x)}{a d-b c}\right )}{m+1}\right )}{b^2 d (m+n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}, x\right ) \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (b x +a \right )^{m} \left (d x +c \right )^{n}\, dx \]
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 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}\,{d x} \]
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 \[ \int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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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