Optimal. Leaf size=60 \[ \frac{(a+b x)^3 \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,2 p+5;p+5;\frac{a+b x}{2 a}\right )}{2 a b (p+4)} \]
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Rubi [A] time = 0.0305929, antiderivative size = 73, normalized size of antiderivative = 1.22, 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} \[ -\frac{a^2 2^{p+3} \left (\frac{b x}{a}+1\right )^{-p-1} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-3,p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
Antiderivative was successfully verified.
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Rule 678
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^3 \left (a^2-b^2 x^2\right )^p \, dx &=\left (a^2 (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 )^{3+p} \, dx\\ &=-\frac{2^{3+p} a^2 \left (1+\frac{b x}{a}\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-3-p,1+p;2+p;\frac{a-b x}{2 a}\right )}{b (1+p)}\\ \end{align*}
Mathematica [B] time = 0.178725, size = 155, normalized size = 2.58 \[ \frac{1}{2} \left (a^2-b^2 x^2\right )^p \left (2 a^3 x \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )+2 a b^2 x^3 \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{b^2 x^2}{a^2}\right )+\frac{\left (b^2 x^2-a^2\right ) \left (a^2 (3 p+7)+b^2 (p+1) x^2\right )}{b (p+1) (p+2)}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.51, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{3} \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 x + a\right )}^{3}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{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^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.8479, size = 476, normalized size = 7.93 \begin{align*} a^{3} a^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )} + 3 a^{2} b \left (\begin{cases} \frac{x^{2} \left (a^{2}\right )^{p}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\begin{cases} \frac{\left (a^{2} - b^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a^{2} - b^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) + a a^{2 p} b^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )} + b^{3} \left (\begin{cases} \frac{x^{4} \left (a^{2}\right )^{p}}{4} & \text{for}\: b = 0 \\- \frac{a^{2} \log{\left (- \frac{a}{b} + x \right )}}{- 2 a^{2} b^{4} + 2 b^{6} x^{2}} - \frac{a^{2} \log{\left (\frac{a}{b} + x \right )}}{- 2 a^{2} b^{4} + 2 b^{6} x^{2}} - \frac{a^{2}}{- 2 a^{2} b^{4} + 2 b^{6} x^{2}} + \frac{b^{2} x^{2} \log{\left (- \frac{a}{b} + x \right )}}{- 2 a^{2} b^{4} + 2 b^{6} x^{2}} + \frac{b^{2} x^{2} \log{\left (\frac{a}{b} + x \right )}}{- 2 a^{2} b^{4} + 2 b^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{a^{2} \log{\left (- \frac{a}{b} + x \right )}}{2 b^{4}} - \frac{a^{2} \log{\left (\frac{a}{b} + x \right )}}{2 b^{4}} - \frac{x^{2}}{2 b^{2}} & \text{for}\: p = -1 \\- \frac{a^{4} \left (a^{2} - b^{2} x^{2}\right )^{p}}{2 b^{4} p^{2} + 6 b^{4} p + 4 b^{4}} - \frac{a^{2} b^{2} p x^{2} \left (a^{2} - b^{2} x^{2}\right )^{p}}{2 b^{4} p^{2} + 6 b^{4} p + 4 b^{4}} + \frac{b^{4} p x^{4} \left (a^{2} - b^{2} x^{2}\right )^{p}}{2 b^{4} p^{2} + 6 b^{4} p + 4 b^{4}} + \frac{b^{4} x^{4} \left (a^{2} - b^{2} x^{2}\right )^{p}}{2 b^{4} p^{2} + 6 b^{4} p + 4 b^{4}} & \text{otherwise} \end{cases}\right ) \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 x + a\right )}^{3}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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