Integrand size = 20, antiderivative size = 209 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=-\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)} \]
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Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {92, 81, 68, 66} \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \]
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Rule 66
Rule 68
Rule 81
Rule 92
Rubi steps \begin{align*} \text {integral}& = \frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\int (b x)^m (c+d x)^n (-b e (c f (1+m)-d e (3+m+n))-b f (c f (2+m)-d e (4+m+n)) x) \, dx}{b d (3+m+n)} \\ & = -\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) \int (b x)^m (c+d x)^n \, dx}{d^2 (2+m+n) (3+m+n)} \\ & = -\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (b x)^m \left (1+\frac {d x}{c}\right )^n \, dx}{d^2 (2+m+n) (3+m+n)} \\ & = -\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.73 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {x (b x)^m (c+d x)^n \left (f (c+d x) (e+f x)-\frac {f (1+m) (c f (2+m)-d e (4+m+n)) (c+d x)+(d e (2+m+n) (c f (1+m)-d e (3+m+n))-c f (1+m) (c f (2+m)-d e (4+m+n))) \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{d (1+m) (2+m+n)}\right )}{d (3+m+n)} \]
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\[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{2}d x\]
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\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
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Result contains complex when optimal does not.
Time = 8.95 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61 \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\frac {b^{m} c^{n} e^{2} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 b^{m} c^{n} e f x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac {b^{m} c^{n} f^{2} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \]
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\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
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\[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int { {\left (f x + e\right )}^{2} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (b x)^m (c+d x)^n (e+f x)^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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