Integrand size = 13, antiderivative size = 46 \[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+\frac {b x}{a}\right )}{b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {69, 67} \[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\frac {2 x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},\frac {b x}{a}+1\right )}{b} \]
[In]
[Out]
Rule 67
Rule 69
Rubi steps \begin{align*} \text {integral}& = \left (x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^m}{\sqrt {a+b x}} \, dx \\ & = \frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+\frac {b x}{a}\right )}{b} \]
[In]
[Out]
\[\int \frac {x^{m}}{\sqrt {b x +a}}d x\]
[In]
[Out]
\[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m}}{\sqrt {b x + a}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\frac {x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 2\right )} \]
[In]
[Out]
\[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m}}{\sqrt {b x + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\int { \frac {x^{m}}{\sqrt {b x + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^m}{\sqrt {a+b x}} \, dx=\int \frac {x^m}{\sqrt {a+b\,x}} \,d x \]
[In]
[Out]