Optimal. Leaf size=70 \[ \frac {(c x)^{m+1} \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} (-m-1),-p;\frac {1-m}{2};-\frac {b}{a x^2}\right )}{c (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {339, 365, 364} \[ \frac {(c x)^{m+1} \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} (-m-1),-p;\frac {1-m}{2};-\frac {b}{a x^2}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 339
Rule 364
Rule 365
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right )^p (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} \left (a+b x^2\right )^p \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {\left (\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \operatorname {Subst}\left (\int x^{-2-m} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} (c x)^{1+m} \, _2F_1\left (\frac {1}{2} (-1-m),-p;\frac {1-m}{2};-\frac {b}{a x^2}\right )}{c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 1.04 \[ \frac {x (c x)^m \left (a+\frac {b}{x^2}\right )^p \left (\frac {a x^2}{b}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} (m-2 p+1),-p;\frac {1}{2} (m-2 p+1)+1;-\frac {a x^2}{b}\right )}{m-2 p+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c x\right )^{m} \left (\frac {a x^{2} + b}{x^{2}}\right )^{p}, x\right ) \]
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 \[ \int \left (c x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{m} \left (a +\frac {b}{x^{2}}\right )^{p}\, dx \]
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 \[ \int \left (c x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,x\right )}^m\,{\left (a+\frac {b}{x^2}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.27, size = 60, normalized size = 0.86 \[ - \frac {a^{p} c^{m} x x^{m} \Gamma \left (- \frac {m}{2} - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - \frac {m}{2} - \frac {1}{2} \\ \frac {1}{2} - \frac {m}{2} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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