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 = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 64
Rule 66
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*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ -\frac {2 (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};-\frac {b x}{a}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 31.96, size = 29, normalized size = 0.67 \[ - \frac {2 a^{n} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - n \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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