Integrand size = 46, antiderivative size = 103 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m,\frac {3}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \]
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Time = 0.05 (sec) , antiderivative size = 103, 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} \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m,\frac {3}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \]
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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 \frac {(a e+c d x)^{-m}}{\sqrt {f+g x}} \, 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 \frac {\left (-\frac {a e g}{c d f-a e g}-\frac {c d g x}{c d f-a e g}\right )^{-m}}{\sqrt {f+g x}} \, dx \\ & = \frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (\frac {1}{2},m;\frac {3}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\frac {2 \left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m ((a e+c d x) (d+e x))^{-m} \sqrt {f+g x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m,\frac {3}{2},\frac {c d (f+g x)}{c d f-a e g}\right )}{g} \]
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\[\int \frac {\left (e x +d \right )^{m} {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{-m}}{\sqrt {g x +f}}d x\]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {g x + f} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {g x + f} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{\sqrt {g x + f} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{\sqrt {f+g x}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{\sqrt {f+g\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
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