Integrand size = 22, antiderivative size = 195 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=-\frac {2 f (9 c f-d e (13+2 n)) (b x)^{7/2} (c+d x)^{1+n}}{b d^2 (9+2 n) (11+2 n)}+\frac {2 f (b x)^{7/2} (c+d x)^{1+n} (e+f x)}{b d (11+2 n)}+\frac {2 \left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) (b x)^{7/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )}{7 b d^2 (9+2 n) (11+2 n)} \]
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Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {92, 81, 68, 66} \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\frac {2 (b x)^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )}{7 b d^2 (2 n+9) (2 n+11)}-\frac {2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d^2 (2 n+9) (2 n+11)}+\frac {2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)} \]
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Rule 66
Rule 68
Rule 81
Rule 92
Rubi steps \begin{align*} \text {integral}& = \frac {2 f (b x)^{7/2} (c+d x)^{1+n} (e+f x)}{b d (11+2 n)}+\frac {2 \int (b x)^{5/2} (c+d x)^n \left (-\frac {1}{2} b e \left (7 c f-2 d e \left (\frac {11}{2}+n\right )\right )-\frac {1}{2} b f (9 c f-d e (13+2 n)) x\right ) \, dx}{b d (11+2 n)} \\ & = -\frac {2 f (9 c f-d e (13+2 n)) (b x)^{7/2} (c+d x)^{1+n}}{b d^2 (9+2 n) (11+2 n)}+\frac {2 f (b x)^{7/2} (c+d x)^{1+n} (e+f x)}{b d (11+2 n)}+\frac {\left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) \int (b x)^{5/2} (c+d x)^n \, dx}{d^2 (9+2 n) (11+2 n)} \\ & = -\frac {2 f (9 c f-d e (13+2 n)) (b x)^{7/2} (c+d x)^{1+n}}{b d^2 (9+2 n) (11+2 n)}+\frac {2 f (b x)^{7/2} (c+d x)^{1+n} (e+f x)}{b d (11+2 n)}+\frac {\left (\left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (b x)^{5/2} \left (1+\frac {d x}{c}\right )^n \, dx}{d^2 (9+2 n) (11+2 n)} \\ & = -\frac {2 f (9 c f-d e (13+2 n)) (b x)^{7/2} (c+d x)^{1+n}}{b d^2 (9+2 n) (11+2 n)}+\frac {2 f (b x)^{7/2} (c+d x)^{1+n} (e+f x)}{b d (11+2 n)}+\frac {2 \left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) (b x)^{7/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (\frac {7}{2},-n;\frac {9}{2};-\frac {d x}{c}\right )}{7 b d^2 (9+2 n) (11+2 n)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.75 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\frac {2 x (b x)^{5/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (-7 f (c+d x) \left (1+\frac {d x}{c}\right )^n (9 c f-d (e (22+4 n)+f (9+2 n) x))+\left (63 c^2 f^2-14 c d e f (11+2 n)+d^2 e^2 \left (99+40 n+4 n^2\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-n,\frac {9}{2},-\frac {d x}{c}\right )\right )}{7 d^2 (9+2 n) (11+2 n)} \]
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\[\int \left (b x \right )^{\frac {5}{2}} \left (d x +c \right )^{n} \left (f x +e \right )^{2}d x\]
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\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \]
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Result contains complex when optimal does not.
Time = 104.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\frac {2 b^{\frac {5}{2}} c^{n} e^{2} x^{\frac {7}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - n \\ \frac {9}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{7} + \frac {4 b^{\frac {5}{2}} c^{n} e f x^{\frac {9}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{2}, - n \\ \frac {11}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{9} + \frac {2 b^{\frac {5}{2}} c^{n} f^{2} x^{\frac {11}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {11}{2}, - n \\ \frac {13}{2} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{11} \]
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\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \]
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\[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int { \left (b x\right )^{\frac {5}{2}} {\left (f x + e\right )}^{2} {\left (d x + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (b x)^{5/2} (c+d x)^n (e+f x)^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^n \,d x \]
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