Optimal. Leaf size=44 \[ \frac {(c x)^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {339, 364} \[ \frac {(c x)^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {1}{2} (-m-1);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 339
Rule 364
Rubi steps
\begin {align*} \int \sqrt {1+\frac {b}{x^2}} (c x)^m \, dx &=-\frac {\left (\left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \operatorname {Subst}\left (\int x^{-2-m} \sqrt {1+b x^2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {(c x)^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{2} (-1-m);\frac {1-m}{2};-\frac {b}{x^2}\right )}{c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.32 \[ \frac {x \sqrt {\frac {b}{x^2}+1} (c x)^m \, _2F_1\left (-\frac {1}{2},\frac {m}{2};\frac {m}{2}+1;-\frac {x^2}{b}\right )}{m \sqrt {\frac {x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c x\right )^{m} \sqrt {\frac {x^{2} + b}{x^{2}}}, x\right ) \]
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 \[ \int \left (c x\right )^{m} \sqrt {\frac {b}{x^{2}} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {b}{x^{2}}+1}\, \left (c x \right )^{m}\, dx \]
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 \[ \int \left (c x\right )^{m} \sqrt {\frac {b}{x^{2}} + 1}\,{d x} \]
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 \[ \int {\left (c\,x\right )}^m\,\sqrt {\frac {b}{x^2}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.04, size = 48, normalized size = 1.09 \[ - \frac {\sqrt {b} c^{m} x^{m} \Gamma \left (- \frac {m}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} \\ \frac {m}{2} + 1 \end {matrix}\middle | {\frac {x^{2} e^{i \pi }}{b}} \right )}}{2 \Gamma \left (1 - \frac {m}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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