Optimal. Leaf size=52 \[ \frac {x^{1-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,\frac {3}{2};\frac {1}{2} (3-2 p);-\frac {b x^2}{a}\right )}{a (1-2 p)} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac {x^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2} (1-2 p),-p;\frac {1}{2} (3-2 p);-\frac {b x^2}{a}\right )}{1-2 p} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int x^{-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^{-2 p} \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {x^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2} (1-2 p),-p;\frac {1}{2} (3-2 p);-\frac {b x^2}{a}\right )}{1-2 p}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 1.25 \[ \frac {x^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2}-p,-p;\frac {3}{2}-p;-\frac {b x^2}{a}\right )}{1-2 p} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p}}{x^{2 \, p}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{x^{2 \, p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int x^{-2 p} \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{x^{2 \, p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x^2+a\right )}^p}{x^{2\,p}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.08, size = 24, normalized size = 0.46 \[ b^{p} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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