Integrand size = 46, antiderivative size = 104 \[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(a e+c d x) \sqrt {d+e x} (f+g x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}+n,2+n,\frac {c d (f+g x)}{c d f-a e g}\right )}{(c d f-a e g) (1+n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13, 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} \[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {d+e x} (f+g x)^n (a e+c d x) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rule 71
Rule 72
Rule 905
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {(f+g x)^n}{\sqrt {a e+c d x}} \, dx}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {\left (\sqrt {a e+c d x} \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n}\right ) \int \frac {\left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^n}{\sqrt {a e+c d x}} \, dx}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {2 (a e+c d x) \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )}{c d \sqrt {d+e x}} \]
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\[\int \frac {\sqrt {e x +d}\, \left (g x +f \right )^{n}}{\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}d x\]
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\[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x} \left (f + g x\right )^{n}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
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\[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,\sqrt {d+e\,x}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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