Optimal. Leaf size=85 \[ -\frac{2^{p+\frac{3}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{3}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
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Rubi [A] time = 0.0638669, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {680, 678, 69} \[ -\frac{2^{p+\frac{3}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{3}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int (a+b x)^{3/2} \left (a^2-b^2 x^2\right )^p \, dx &=\frac{\left (a \sqrt{a+b x}\right ) \int \left (1+\frac{b x}{a}\right )^{3/2} \left (a^2-b^2 x^2\right )^p \, dx}{\sqrt{1+\frac{b x}{a}}}\\ &=\left (a \sqrt{a+b x} \left (1+\frac{b x}{a}\right )^{-\frac{3}{2}-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 )^{\frac{3}{2}+p} \left (a^2-a b x\right )^p \, dx\\ &=-\frac{2^{\frac{3}{2}+p} \sqrt{a+b x} \left (1+\frac{b x}{a}\right )^{-\frac{3}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-\frac{3}{2}-p,1+p;2+p;\frac{a-b x}{2 a}\right )}{b (1+p)}\\ \end{align*}
Mathematica [C] time = 0.256433, size = 189, normalized size = 2.22 \[ \frac{2^{p-1} \sqrt{a+b x} \left (1-\frac{b x}{a}\right )^{-p} \left (\frac{b x}{a}+1\right )^{-2 p-\frac{1}{2}} \left (b^2 (p+1) x^2 (a-b x)^p (a+b x)^p \left (\frac{b x}{2 a}+\frac{1}{2}\right )^p F_1\left (2;-p,-p-\frac{1}{2};3;\frac{b x}{a},-\frac{b x}{a}\right )-2 \sqrt{2} a (a-b x) \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^p \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )\right )}{b (p+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.514, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{3}{2}}} \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 )}^{\frac{3}{2}}{\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 x + a\right )}^{\frac{3}{2}}{\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{\frac{3}{2}}{\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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