Integrand size = 19, antiderivative size = 67 \[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\frac {2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \sqrt {b x+c x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},1+\frac {c x}{b}\right )}{3 c} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {688, 69, 67} \[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\frac {2 (b+c x) \sqrt {b x+c x^2} (d x)^m \left (-\frac {c x}{b}\right )^{-m-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m-\frac {1}{2},\frac {5}{2},\frac {c x}{b}+1\right )}{3 c} \]
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Rule 67
Rule 69
Rule 688
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-\frac {1}{2}-m} (d x)^m \sqrt {b x+c x^2}\right ) \int x^{\frac {1}{2}+m} \sqrt {b+c x} \, dx}{\sqrt {b+c x}} \\ & = \frac {\left (\left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m \sqrt {b x+c x^2}\right ) \int \left (-\frac {c x}{b}\right )^{\frac {1}{2}+m} \sqrt {b+c x} \, dx}{\sqrt {b+c x}} \\ & = \frac {2 \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \sqrt {b x+c x^2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};1+\frac {c x}{b}\right )}{3 c} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int (d x)^m \sqrt {b x+c x^2} \, dx=-\frac {2 \left (-\frac {c x}{b}\right )^{-\frac {3}{2}-m} (d x)^m (x (b+c x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-m,\frac {5}{2},1+\frac {c x}{b}\right )}{3 b} \]
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\[\int \left (d x \right )^{m} \sqrt {c \,x^{2}+b x}d x\]
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\[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\int \left (d x\right )^{m} \sqrt {x \left (b + c x\right )}\, dx \]
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\[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x} \left (d x\right )^{m} \,d x } \]
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Timed out. \[ \int (d x)^m \sqrt {b x+c x^2} \, dx=\int \sqrt {c\,x^2+b\,x}\,{\left (d\,x\right )}^m \,d x \]
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