Integrand size = 27, antiderivative size = 256 \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\frac {(b g-a h) (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (1+m)}+\frac {h (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (2+m,-n,-p,3+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (2+m)} \]
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Time = 0.14 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {165, 145, 144, 143} \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\frac {(b g-a h) (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+1)}+\frac {h (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+2,-n,-p,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+2)} \]
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Rule 143
Rule 144
Rule 145
Rule 165
Rubi steps \begin{align*} \text {integral}& = \frac {h \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b}+\frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p \, dx}{b} \\ & = \frac {\left (h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b}+\frac {\left ((b g-a h) (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 (e+f x)^p \, dx}{b} \\ & = \frac {\left (h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b}+\frac {\left ((b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b} \\ & = \frac {(b g-a h) (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (1+m)}+\frac {h (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (2+m;-n,-p;3+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (2+m)} \\ \end{align*}
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx \]
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\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p} \left (h x +g \right )d x\]
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\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\text {Timed out} \]
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\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\int { {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx=\int {\left (e+f\,x\right )}^p\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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