Optimal. Leaf size=58 \[ -\frac {\left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,2 p;p;\frac {a+b x}{2 a}\right )}{2 a b (1-p) (a+b x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {678, 69} \[ -\frac {2^{p-2} \left (\frac {b x}{a}+1\right )^{-p-1} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac {a-b x}{2 a}\right )}{a^3 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (a^2-b^2 x^2\right )^p}{(a+b x)^2} \, dx &=\frac {\left ((a-b x)^{-1-p} \left (1+\frac {b x}{a}\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p}\right ) \int (a-b x)^p \left (1+\frac {b x}{a}\right )^{-2+p} \, dx}{a^3}\\ &=-\frac {2^{-2+p} \left (1+\frac {b x}{a}\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (2-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{a^3 b (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.29 \[ -\frac {2^{p-2} (a-b x) \left (\frac {b x}{a}+1\right )^{-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac {a-b x}{2 a}\right )}{a^2 b (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b^{2} x^{2}+a^{2}\right )^{p}}{\left (b x +a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a^2-b^2\,x^2\right )}^p}{{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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