Integrand size = 29, antiderivative size = 245 \[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{2-m} (4 b d (f g+e h)-a d f h (3-m)-b c f h (2+m)+3 b d f h x)}{12 b^2 d^2}+\frac {(b c-a d) \left (a^2 d^2 f h \left (6-5 m+m^2\right )-2 a b d (2-m) (2 d (f g+e h)-c f h (1+m))+b^2 \left (12 d^2 e g-4 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-1+m,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (1+m)} \]
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Time = 0.09 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {152, 72, 71} \[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m-1,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h \left (m^2-5 m+6\right )-2 a b d (2-m) (2 d (e h+f g)-c f h (m+1))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )-4 c d (m+1) (e h+f g)+12 d^2 e g\right )\right )}{12 b^4 d^2 (m+1)}+\frac {(a+b x)^{m+1} (c+d x)^{2-m} (-a d f h (3-m)-b c f h (m+2)+4 b d (e h+f g)+3 b d f h x)}{12 b^2 d^2} \]
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Rule 71
Rule 72
Rule 152
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+m} (c+d x)^{2-m} (4 b d (f g+e h)-a d f h (3-m)-b c f h (2+m)+3 b d f h x)}{12 b^2 d^2}+\frac {\left (a^2 d^2 f h \left (6-5 m+m^2\right )-2 a b d (2-m) (2 d (f g+e h)-c f h (1+m))+b^2 \left (12 d^2 e g-4 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{1-m} \, dx}{12 b^2 d^2} \\ & = \frac {(a+b x)^{1+m} (c+d x)^{2-m} (4 b d (f g+e h)-a d f h (3-m)-b c f h (2+m)+3 b d f h x)}{12 b^2 d^2}+\frac {\left ((b c-a d) \left (a^2 d^2 f h \left (6-5 m+m^2\right )-2 a b d (2-m) (2 d (f g+e h)-c f h (1+m))+b^2 \left (12 d^2 e g-4 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{1-m} \, dx}{12 b^3 d^2} \\ & = \frac {(a+b x)^{1+m} (c+d x)^{2-m} (4 b d (f g+e h)-a d f h (3-m)-b c f h (2+m)+3 b d f h x)}{12 b^2 d^2}+\frac {(b c-a d) \left (a^2 d^2 f h \left (6-5 m+m^2\right )-2 a b d (2-m) (2 d (f g+e h)-c f h (1+m))+b^2 \left (12 d^2 e g-4 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-1+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (1+m)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.80 \[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{1-m} \left (\frac {b (c+d x)}{b c-a d}\right )^{-1+m} \left ((b c-a d)^2 f h \operatorname {Hypergeometric2F1}\left (-3+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )+b \left (-\left ((b c-a d) (2 c f h-d (f g+e h)) \operatorname {Hypergeometric2F1}\left (-2+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )\right )+b (d e-c f) (d g-c h) \operatorname {Hypergeometric2F1}\left (-1+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3 d^2 (1+m)} \]
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\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{1-m} \left (f x +e \right ) \left (h x +g \right )d x\]
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\[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\int { {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1} \,d x } \]
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Exception generated. \[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\int { {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1} \,d x } \]
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\[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\int { {\left (f x + e\right )} {\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^{1-m} (e+f x) (g+h x) \, dx=\int \left (e+f\,x\right )\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m} \,d x \]
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