Optimal. Leaf size=43 \[ -\frac {2 (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-\frac {1}{2},-n;\frac {1}{2};-\frac {b x}{a}\right )}{\sqrt {x}} \]
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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 = {68, 66}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
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 = 0.09, size = 43, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 15.30, size = 26, normalized size = 0.60 \begin {gather*} \frac {-2 a^n \text {hyper}\left [\left \{-\frac {1}{2},-n\right \},\left \{\frac {1}{2}\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{\sqrt {x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 14.71, size = 29, normalized size = 0.67 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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