Optimal. Leaf size=53 \[ -\frac {x^{-2 p-3} \left (a+b x^2\right )^{p+1} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2} (-2 p-1);-\frac {b x^2}{a}\right )}{a (2 p+3)} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.32, 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 = {365, 364} \[ -\frac {x^{-2 p-3} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} (-2 p-3),-p;\frac {1}{2} (-2 p-1);-\frac {b x^2}{a}\right )}{2 p+3} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int x^{-4-2 p} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^{-4-2 p} \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=-\frac {x^{-3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} (-3-2 p),-p;\frac {1}{2} (-1-2 p);-\frac {b x^2}{a}\right )}{3+2 p}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 1.25 \[ -\frac {x^{-2 p-3} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-p-\frac {3}{2},-p;-p-\frac {1}{2};-\frac {b x^2}{a}\right )}{2 p+3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 4}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int x^{-2 p -4} \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (b x^{2} + a\right )}^{p} x^{-2 \, p - 4}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^p}{x^{2\,p+4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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