Optimal. Leaf size=30 \[ -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \]
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Rubi [A]
time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {67}
\begin {gather*} -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rubi steps
\begin {align*} \int \frac {x^n}{\sqrt {a-a x}} \, dx &=-\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \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 = 2.19, size = 26, normalized size = 0.87 \begin {gather*} \frac {-2 I \sqrt {-1+x} \text {hyper}\left [\left \{\frac {1}{2},-n\right \},\left \{\frac {3}{2}\right \},\left (-1+x\right ) \text {exp\_polar}\left [I \text {Pi}\right ]\right ]}{\sqrt {a}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{n}}{\sqrt {-a x +a}}\, dx\]
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 = 0.55, size = 31, normalized size = 1.03 \begin {gather*} - \frac {2 i \sqrt {x - 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\left (x - 1\right ) e^{i \pi }} \right )}}{\sqrt {a}} \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 = -0.03 \begin {gather*} \int \frac {x^n}{\sqrt {a-a\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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