Integrand size = 29, antiderivative size = 530 \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\frac {(b g-a h)^3 (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (1+m)}+\frac {3 h (b g-a h)^2 (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (2+m,-n,-p,3+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (2+m)}+\frac {3 h^2 (b g-a h) (a+b x)^{3+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (3+m,-n,-p,4+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}+\frac {h^3 (a+b x)^{4+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (4+m,-n,-p,5+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (4+m)} \]
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Time = 0.80 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {187, 165, 145, 144, 143} \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\frac {3 h^2 (b g-a h) (a+b x)^{m+3} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+3,-n,-p,m+4,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+3)}+\frac {(b g-a h)^3 (a+b x)^{m+1} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+1)}+\frac {3 h (b g-a h)^2 (a+b x)^{m+2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+2,-n,-p,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+2)}+\frac {h^3 (a+b x)^{m+4} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \operatorname {AppellF1}\left (m+4,-n,-p,m+5,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (m+4)} \]
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Rule 143
Rule 144
Rule 145
Rule 165
Rule 187
Rubi steps \begin{align*} \text {integral}& = \frac {h \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b}+\frac {(b g-a h) \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^2 \, dx}{b} \\ & = \frac {h^2 \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+2 \frac {(h (b g-a h)) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2}+\frac {(b g-a h)^2 \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x) \, dx}{b^2} \\ & = \frac {h^3 \int (a+b x)^{3+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+2 \left (\frac {\left (h^2 (b g-a h)\right ) \int (a+b x)^{2+m} (c+d x)^n (e+f x)^p \, dx}{b^3}+\frac {\left (h (b g-a h)^2\right ) \int (a+b x)^{1+m} (c+d x)^n (e+f x)^p \, dx}{b^3}\right )+\frac {(b g-a h)^3 \int (a+b x)^m (c+d x)^n (e+f x)^p \, dx}{b^3} \\ & = \frac {\left (h^3 (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{3+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^3}+\frac {\left (h^2 (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)^{2+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^3}+\frac {\left (h (b g-a h)^2 (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^3}+2 \left (\frac {\left (h^2 (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)^{2+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^3}+\frac {\left (h (b g-a h)^2 (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^3}\right )+\frac {\left ((b g-a h)^3 (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^3} \\ & = \frac {\left (h^3 (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)^{3+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^3}+\frac {\left (h^2 (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)^{2+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^3}+\frac {\left (h (b g-a h)^2 (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^3}+2 \left (\frac {\left (h^2 (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)^{2+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^3}+\frac {\left (h (b g-a h)^2 (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^3}\right )+\frac {\left ((b g-a h)^3 (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^3} \\ & = \frac {(b g-a h)^3 (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^4 (1+m)}+\frac {h (b g-a h)^2 (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^4 (2+m)}+\frac {h^2 (b g-a h) (a+b x)^{3+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 (3+m;-n,-p;4+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}+2 \left (\frac {h (b g-a h)^2 (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^4 (2+m)}+\frac {h^2 (b g-a h) (a+b x)^{3+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 (3+m;-n,-p;4+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (3+m)}\right )+\frac {h^3 (a+b x)^{4+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 (4+m;-n,-p;5+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^4 (4+m)} \\ \end{align*}
\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx \]
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\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p} \left (h x +g \right )^{3}d x\]
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\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\text {Timed out} \]
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\[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int { {\left (h x + g\right )}^{3} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\text {Timed out} \]
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Timed out. \[ \int (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^3 \, dx=\int {\left (e+f\,x\right )}^p\,{\left (g+h\,x\right )}^3\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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