Integrand size = 46, antiderivative size = 213 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}+\frac {\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) (a e+c d x) \sqrt {d+e x} (f+g x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}+n,2+n,\frac {c d (f+g x)}{c d f-a e g}\right )}{c d g (c d f-a e g) (1+n) (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {894, 905, 72, 71} \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 e (f+g x)^{n+1} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (f+g x)^n (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (2 n+3) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rule 71
Rule 72
Rule 894
Rule 905
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \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}{c d g (3+2 n)} \\ & = \frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {(f+g x)^n}{\sqrt {a e+c d x}} \, dx}{c d g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \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}{c d g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {2 \left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) (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^2 d^2 g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} (f+g x)^n \left (c d e (f+g x)+\left (-2 a e^2 g (1+n)+c d (-e f+d g (3+2 n))\right ) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {g (a e+c d x)}{-c d f+a e g}\right )\right )}{c^2 d^2 g \left (\frac {3}{2}+n\right ) \sqrt {d+e x}} \]
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\[\int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{n}}{\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}d x\]
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\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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\[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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