Integrand size = 46, antiderivative size = 105 \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {905, 72, 71} \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
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Rule 71
Rule 72
Rule 905
Rubi steps \begin{align*} \text {integral}& = \left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (a e+c d x)^{-m} (f+g x)^{3/2} \, dx \\ & = \left (\left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (f+g x)^{3/2} \left (-\frac {a e g}{c d f-a e g}-\frac {c d g x}{c d f-a e g}\right )^{-m} \, dx \\ & = \frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (\frac {5}{2},m;\frac {7}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\frac {2 \left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} (f+g x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},m,\frac {7}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
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\[\int \left (e x +d \right )^{m} \left (g x +f \right )^{\frac {3}{2}} {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{-m}d x\]
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\[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\text {Timed out} \]
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\[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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\[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Timed out. \[ \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^m}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
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