Integrand size = 15, antiderivative size = 48 \[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-m,\frac {3}{2},1+\frac {b x}{a}\right )}{a} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {69, 67} \[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-m,\frac {3}{2},\frac {b x}{a}+1\right )}{a} \]
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Rule 67
Rule 69
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^{-1+m}}{\sqrt {a+b x}} \, dx}{a} \\ & = -\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},1-m;\frac {3}{2};1+\frac {b x}{a}\right )}{a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-m,\frac {3}{2},1+\frac {b x}{a}\right )}{a} \]
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\[\int \frac {x^{-1+m}}{\sqrt {b x +a}}d x\]
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\[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m - 1}}{\sqrt {b x + a}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=\frac {a^{m} a^{- m - \frac {1}{2}} x^{m} \Gamma \left (m\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m \\ m + 1 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 1\right )} \]
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\[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m - 1}}{\sqrt {b x + a}} \,d x } \]
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\[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m - 1}}{\sqrt {b x + a}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+m}}{\sqrt {a+b x}} \, dx=\int \frac {x^{m-1}}{\sqrt {a+b\,x}} \,d x \]
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