Optimal. Leaf size=63 \[ \frac{(a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,m+2 p+2;m+p+2;\frac{a+b x}{2 a}\right )}{2 a b (m+p+1)} \]
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Rubi [A] time = 0.055875, antiderivative size = 85, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {680, 678, 69} \[ -\frac{2^{m+p} (a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \left (\frac{b x}{a}+1\right )^{-m-p-1} \, _2F_1\left (-m-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx &=\left ((a+b x)^m \left (1+\frac{b x}{a}\right )^{-m}\right ) \int \left (1+\frac{b x}{a}\right )^m \left (a^2-b^2 x^2\right )^p \, dx\\ &=\left ((a+b x)^m \left (1+\frac{b x}{a}\right )^{-1-m-p} \left (a^2-a b x\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p}\right ) \int \left (1+\frac{b x}{a}\right )^{m+p} \left (a^2-a b x\right )^p \, dx\\ &=-\frac{2^{m+p} (a+b x)^m \left (1+\frac{b x}{a}\right )^{-1-m-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-m-p,1+p;2+p;\frac{a-b x}{2 a}\right )}{a b (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0899089, size = 85, normalized size = 1.35 \[ \frac{2^{m+p} (b x-a) (a+b x)^m \left (a^2-b^2 x^2\right )^p \left (\frac{b x}{a}+1\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.625, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, 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{\left (-b^{2} x^{2} + a^{2}\right )}^{p}{\left (b x + a\right )}^{m}\,{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 ({\left (-b^{2} x^{2} + a^{2}\right )}^{p}{\left (b x + a\right )}^{m}, 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 \left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p} \left (a + b x\right )^{m}\, 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{\left (-b^{2} x^{2} + a^{2}\right )}^{p}{\left (b x + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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