Integrand size = 13, antiderivative size = 43 \[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=-\frac {2 (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},-\frac {b x}{a}\right )}{\sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 43, 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 = {68, 66} \[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=-\frac {2 (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},-\frac {b x}{a}\right )}{\sqrt {x}} \]
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Rule 66
Rule 68
Rubi steps \begin{align*} \text {integral}& = \left ((a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int \frac {\left (1+\frac {b x}{a}\right )^n}{x^{3/2}} \, dx \\ & = -\frac {2 (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};-\frac {b x}{a}\right )}{\sqrt {x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=-\frac {2 (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},-\frac {b x}{a}\right )}{\sqrt {x}} \]
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\[\int \frac {\left (b x +a \right )^{n}}{x^{\frac {3}{2}}}d x\]
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\[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 22.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=- \frac {2 a^{n} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - n \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {x}} \]
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\[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^n}{x^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^{3/2}} \,d x \]
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