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 = 0.0309562, 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 678
Rule 69
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*}
Mathematica [A] time = 0.0531989, 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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Maple [F] time = 0.541, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p}}{\left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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