Integrand size = 22, antiderivative size = 53 \[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (d x)^m (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (m,-n,1+m,-\frac {b x}{a}\right )}{m \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 16, 68, 66} \[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (m,-n,m+1,-\frac {b x}{a}\right )}{m \sqrt {c x^2}} \]
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Rule 15
Rule 16
Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(d x)^m (a+b x)^n}{x} \, dx}{\sqrt {c x^2}} \\ & = \frac {(d x) \int (d x)^{-1+m} (a+b x)^n \, dx}{\sqrt {c x^2}} \\ & = \frac {\left (d x (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{-1+m} \left (1+\frac {b x}{a}\right )^n \, dx}{\sqrt {c x^2}} \\ & = \frac {x (d x)^m (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (m,-n;1+m;-\frac {b x}{a}\right )}{m \sqrt {c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (d x)^m (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (m,-n,1+m,-\frac {b x}{a}\right )}{m \sqrt {c x^2}} \]
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\[\int \frac {\left (d x \right )^{m} \left (b x +a \right )^{n}}{\sqrt {c \,x^{2}}}d x\]
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\[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\sqrt {c x^{2}}} \,d x } \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\int \frac {\left (d x\right )^{m} \left (a + b x\right )^{n}}{\sqrt {c x^{2}}}\, dx \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\sqrt {c x^{2}}} \,d x } \]
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\[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\sqrt {c x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m (a+b x)^n}{\sqrt {c x^2}} \, dx=\int \frac {{\left (d\,x\right )}^m\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \,d x \]
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