Optimal. Leaf size=109 \[ \frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac {\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}} x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2-m);-\frac {m}{2};\frac {a^2}{x^2}\right )}{a \left (2+3 m+m^2\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6470, 30, 265,
346, 371} \begin {gather*} -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-2);-\frac {m}{2};\frac {a^2}{x^2}\right )}{a \left (m^2+3 m+2\right )}-\frac {x^{m+2}}{a \left (m^2+3 m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 265
Rule 346
Rule 371
Rule 6470
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^m \, dx &=\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {\int x^{1+m} \, dx}{a (1+m)}-\frac {\left (\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}}\right ) \int \frac {x^{1+m}}{\sqrt {1-\frac {a}{x}} \sqrt {1+\frac {a}{x}}} \, dx}{a (1+m)}\\ &=\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac {\left (\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}}\right ) \int \frac {x^{1+m}}{\sqrt {1-\frac {a^2}{x^2}}} \, dx}{a (1+m)}\\ &=\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {x^{2+m}}{a \left (2+3 m+m^2\right )}+\frac {\left (\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}} \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-3-m}}{\sqrt {1-a^2 x^2}} \, dx,x,\frac {1}{x}\right )}{a (1+m)}\\ &=\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} x^{1+m}}{1+m}-\frac {x^{2+m}}{a \left (2+3 m+m^2\right )}-\frac {\sqrt {\frac {1}{1+\frac {a}{x}}} \sqrt {1+\frac {a}{x}} x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2-m);-\frac {m}{2};\frac {a^2}{x^2}\right )}{a \left (2+3 m+m^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 147, normalized size = 1.35 \begin {gather*} -\frac {2^{-1-m} a \left (\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{1+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}}\right )^{-m} \left (1+e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )^{-m} \left (\frac {a}{x}\right )^m x^m \left (e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )} m \, _2F_1\left (1-\frac {m}{2},-m;2-\frac {m}{2};-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )-(-2+m) \, _2F_1\left (-m,-\frac {m}{2};1-\frac {m}{2};-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )}{(-2+m) m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (\frac {x}{a}+\sqrt {\frac {x}{a}-1}\, \sqrt {\frac {x}{a}+1}\right ) x^{m}\, dx\]
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 = 0.00, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int x x^{m}\, dx + \int a x^{m} \sqrt {-1 + \frac {x}{a}} \sqrt {1 + \frac {x}{a}}\, dx}{a} \end {gather*}
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 \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 = -0.01 \begin {gather*} \int x^m\,\left (\sqrt {\frac {x}{a}-1}\,\sqrt {\frac {x}{a}+1}+\frac {x}{a}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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