Optimal. Leaf size=43 \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{1}{2},-n;\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0086445, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {66, 64} \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{1}{2},-n;\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 66
Rule 64
Rubi steps
\begin{align*} \int \frac{(a+b x)^n}{x^{3/2}} \, dx &=\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*}
Mathematica [A] time = 0.0072898, size = 43, normalized size = 1. \[ -\frac{2 (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac{1}{2},-n;\frac{1}{2};-\frac{b x}{a}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{n}{x}^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 54.5904, size = 29, normalized size = 0.67 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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