Integrand size = 15, antiderivative size = 49 \[ \int \frac {x^{1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 a x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-1-m,\frac {3}{2},1+\frac {b x}{a}\right )}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int \frac {x^{1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 a x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m-1,\frac {3}{2},\frac {b x}{a}+1\right )}{b^2} \]
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Rule 67
Rule 69
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a 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}{b} \\ & = -\frac {2 a 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 )}{b^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {x^{1+m}}{\sqrt {a+b x}} \, dx=-\frac {2 a x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-1-m,\frac {3}{2},1+\frac {b x}{a}\right )}{b^2} \]
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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 = 1.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \frac {x^{1+m}}{\sqrt {a+b x}} \, dx=\frac {x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 3\right )} \]
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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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