Integrand size = 16, antiderivative size = 117 \[ \int x (a+b x)^n (c+d x)^p \, dx=\frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}+\frac {(b c (1+n)+a d (1+p)) (a+b x)^{1+n} (c+d x)^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+n+p,2+p,\frac {b (c+d x)}{b c-a d}\right )}{b d (b c-a d) (1+p) (2+n+p)} \]
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Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {81, 72, 71} \[ \int x (a+b x)^n (c+d x)^p \, dx=\frac {(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)}-\frac {(a+b x)^{n+1} (c+d x)^p (a d (p+1)+b c (n+1)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (n+1,-p,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (n+1) (n+p+2)} \]
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Rule 71
Rule 72
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {(b c (1+n)+a d (1+p)) \int (a+b x)^n (c+d x)^p \, dx}{b d (2+n+p)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {\left ((b c (1+n)+a d (1+p)) (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p}\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p \, dx}{b d (2+n+p)} \\ & = \frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {(b c (1+n)+a d (1+p)) (a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (1+n,-p;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (1+n) (2+n+p)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90 \[ \int x (a+b x)^n (c+d x)^p \, dx=\frac {(a+b x)^{1+n} (c+d x)^p \left (b (c+d x)-\frac {(b c (1+n)+a d (1+p)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+n,-p,2+n,\frac {d (a+b x)}{-b c+a d}\right )}{1+n}\right )}{b^2 d (2+n+p)} \]
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\[\int x \left (b x +a \right )^{n} \left (d x +c \right )^{p}d x\]
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\[ \int x (a+b x)^n (c+d x)^p \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x \,d x } \]
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Exception generated. \[ \int x (a+b x)^n (c+d x)^p \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int x (a+b x)^n (c+d x)^p \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x \,d x } \]
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\[ \int x (a+b x)^n (c+d x)^p \, dx=\int { {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x \,d x } \]
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Timed out. \[ \int x (a+b x)^n (c+d x)^p \, dx=\int x\,{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^p \,d x \]
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