Optimal. Leaf size=109 \[ -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-2);-\frac {m}{2};\frac {a^2}{x^2}\right )}{a \left (m^2+3 m+2\right )}-\frac {x^{m+2}}{a \left (m^2+3 m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1} \]
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Rubi [A] time = 0.08, 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 = {6335, 30, 259, 339, 364} \[ -\frac {\sqrt {\frac {1}{\frac {a}{x}+1}} \sqrt {\frac {a}{x}+1} x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-m-2);-\frac {m}{2};\frac {a^2}{x^2}\right )}{a \left (m^2+3 m+2\right )}-\frac {x^{m+2}}{a \left (m^2+3 m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{m+1} \]
Antiderivative was successfully verified.
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Rule 30
Rule 259
Rule 339
Rule 364
Rule 6335
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 ) \operatorname {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 = 0.84, size = 139, normalized size = 1.28 \[ -\frac {a 2^{-m-1} x^m \left (\frac {a}{x}\right )^m e^{\text {sech}^{-1}\left (\frac {a}{x}\right )} \left (\frac {e^{\text {sech}^{-1}\left (\frac {a}{x}\right )}}{e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}+1}\right )^{-m-1} \left (m e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )} \, _2F_1\left (1,\frac {m}{2}+2;2-\frac {m}{2};-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )-(m-2) \, _2F_1\left (1,\frac {m}{2}+1;1-\frac {m}{2};-e^{2 \text {sech}^{-1}\left (\frac {a}{x}\right )}\right )\right )}{(m-2) m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{m} \sqrt {\frac {a + x}{a}} \sqrt {-\frac {a - x}{a}} + x x^{m}}{a}, x\right ) \]
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 \[ \int x^{m} {\left (\sqrt {\frac {x}{a} + 1} \sqrt {\frac {x}{a} - 1} + \frac {x}{a}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \left (\frac {x}{a}+\sqrt {-1+\frac {x}{a}}\, \sqrt {1+\frac {x}{a}}\right ) x^{m}\, dx \]
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 \[ \frac {x^{2} x^{m}}{a {\left (m + 2\right )}} + \frac {\int \sqrt {a + x} \sqrt {-a + x} x^{m}\,{d x}}{a} \]
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 \[ \int x^m\,\left (\sqrt {\frac {x}{a}-1}\,\sqrt {\frac {x}{a}+1}+\frac {x}{a}\right ) \,d x \]
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 \[ \frac {\int x x^{m}\, dx + \int a x^{m} \sqrt {-1 + \frac {x}{a}} \sqrt {1 + \frac {x}{a}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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