Integrand size = 46, antiderivative size = 104 \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=-\frac {(a e+c d x) (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+n,2+n,\frac {c d (f+g x)}{c d f-a e g}\right )}{(c d f-a e g) (1+n) \sqrt {d+e x}} \]
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Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.15, 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} \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 (f+g x)^n (a e+c d x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{3 c d \sqrt {d+e x}} \]
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Rule 71
Rule 72
Rule 905
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \int \sqrt {a e+c d x} (f+g x)^n \, dx}{\sqrt {a e+c d x} \sqrt {d+e x}} \\ & = \frac {\left ((f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}\right ) \int \sqrt {a e+c d x} \left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^n \, dx}{\sqrt {a e+c d x} \sqrt {d+e x}} \\ & = \frac {2 (a e+c d x) (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{3 c d \sqrt {d+e x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )}{3 c d (d+e x)^{3/2}} \]
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\[\int \frac {\left (g x +f \right )^{n} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{\sqrt {e x +d}}d x\]
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\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{n}}{\sqrt {d + e x}}\, dx \]
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\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x)^n \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\sqrt {d+e\,x}} \,d x \]
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