Integrand size = 60, antiderivative size = 93 \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{e n \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \]
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Time = 0.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2152, 2150, 2146, 12, 30} \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \]
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Rule 12
Rule 30
Rule 2146
Rule 2150
Rule 2152
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \, dx \\ & = \frac {\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}} \, dx}{\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \\ & = \frac {\left (2 f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {x^{-1+n}}{2 e} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \\ & = \frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right ) \text {Subst}\left (\int x^{-1+n} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{e \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \\ & = \frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{e n \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\frac {f \sqrt {a+\frac {e x (2 d+e x)}{f^2}} \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^n}{e n \sqrt {g \left (a+\frac {e x (2 d+e x)}{f^2}\right )}} \]
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\[\int \frac {{\left (d +e x +f \sqrt {\frac {a \,f^{2}+e x \left (e x +2 d \right )}{f^{2}}}\right )}^{n}}{\sqrt {\frac {a \,f^{2} g +e g x \left (e x +2 d \right )}{f^{2}}}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\frac {{\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f^{3} \sqrt {\frac {e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}} \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}}{e^{3} g n x^{2} + a e f^{2} g n + 2 \, d e^{2} g n x} \]
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Timed out. \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\int { \frac {{\left (e x + d + \sqrt {a f^{2} + {\left (e x + 2 \, d\right )} e x}\right )}^{n}}{\frac {\sqrt {a f^{2} g + {\left (e x + 2 \, d\right )} e g x}}{f}} \,d x } \]
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\[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\int { \frac {{\left (e x + f \sqrt {\frac {a f^{2} + {\left (e x + 2 \, d\right )} e x}{f^{2}}} + d\right )}^{n}}{\sqrt {\frac {a f^{2} g + {\left (e x + 2 \, d\right )} e g x}{f^{2}}}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2 g+e g x (2 d+e x)}{f^2}}} \, dx=\int \frac {{\left (d+e\,x+f\,\sqrt {\frac {a\,f^2+e\,x\,\left (2\,d+e\,x\right )}{f^2}}\right )}^n}{\sqrt {\frac {a\,g\,f^2+e\,g\,x\,\left (2\,d+e\,x\right )}{f^2}}} \,d x \]
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