Integrand size = 44, antiderivative size = 221 \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}+\frac {(2 c d-b e)^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} (c d-b e-c e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (7+m)} \]
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Time = 0.23 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {808, 693, 691, 72, 71} \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m-\frac {5}{2},\frac {9}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]
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Rule 71
Rule 72
Rule 691
Rule 693
Rule 808
Rubi steps \begin{align*} \text {integral}& = -\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac {(b e g (7+2 m)-2 c (d g m+e f (7+m))) \int (d+e x)^m \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{2 c e (7+m)} \\ & = -\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac {\left ((b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \left (1+\frac {e x}{d}\right )^m \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{2 c e (7+m)} \\ & = -\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac {\left ((b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-m} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {5}{2}+m} \left (c d^2-b d e-c d e x\right )^{5/2} \, dx}{2 c e (7+m) \sqrt {c d^2-b d e-c d e x}} \\ & = -\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}-\frac {\left (\left (-c d e-\frac {e \left (c d^2-b d e\right )}{d}\right )^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (-\frac {c d e \left (1+\frac {e x}{d}\right )}{-c d e-\frac {e \left (c d^2-b d e\right )}{d}}\right )^{-\frac {1}{2}-m} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}\right ) \int \left (c d^2-b d e-c d e x\right )^{5/2} \left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{\frac {5}{2}+m} \, dx}{2 c^3 d^2 e^3 (7+m) \sqrt {c d^2-b d e-c d e x}} \\ & = -\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (7+m)}+\frac {(2 c d-b e)^2 (b e g (7+2 m)-2 c (d g m+e f (7+m))) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} (c d-b e-c e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \, _2F_1\left (\frac {7}{2},-\frac {5}{2}-m;\frac {9}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (7+m)} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.73 \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x)^{-3+m} ((d+e x) (-b e+c (d-e x)))^{7/2} \left (-7 c^3 g (d+e x)^3-(-2 c d+b e)^2 (-b e g (7+2 m)+2 c (d g m+e f (7+m))) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {-c d+b e+c e x}{-2 c d+b e}\right )\right )}{7 c^4 e^2 (7+m)} \]
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\[\int \left (e x +d \right )^{m} \left (g x +f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}d x\]
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\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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Exception generated. \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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\[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int { {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )} {\left (e x + d\right )}^{m} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2} \,d x \]
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