Optimal. Leaf size=45 \[ \frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 45, 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}{3} x^{3/2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rubi steps
\begin {align*} \int \sqrt {x} (a+b x)^n \, dx &=\left ((a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int \sqrt {x} \left (1+\frac {b x}{a}\right )^n \, dx\\ &=\frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \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 = 5.18, size = 26, normalized size = 0.58 \begin {gather*} \frac {2 x^{\frac {3}{2}} a^n \text {hyper}\left [\left \{\frac {3}{2},-n\right \},\left \{\frac {5}{2}\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {x}\, \left (b x +a \right )^{n}\, dx\]
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 = 3.77, size = 27, normalized size = 0.60 \begin {gather*} \frac {2 a^{n} x^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - n \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{3} \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 \sqrt {x}\,{\left (a+b\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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