Optimal. Leaf size=88 \[ -\frac{2^{p-\frac{1}{2}} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1) \sqrt{a+b x}} \]
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Rubi [A] time = 0.0666792, antiderivative size = 88, 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} \[ -\frac{2^{p-\frac{1}{2}} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \frac{\left (a^2-b^2 x^2\right )^p}{\sqrt{a+b x}} \, dx &=\frac{\sqrt{1+\frac{b x}{a}} \int \frac{\left (a^2-b^2 x^2\right )^p}{\sqrt{1+\frac{b x}{a}}} \, dx}{\sqrt{a+b x}}\\ &=\frac{\left (\left (1+\frac{b x}{a}\right )^{-\frac{1}{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{1}{2}+p} \left (a^2-a b x\right )^p \, dx}{\sqrt{a+b x}}\\ &=-\frac{2^{-\frac{1}{2}+p} \left (1+\frac{b x}{a}\right )^{-\frac{1}{2}-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (\frac{1}{2}-p,1+p;2+p;\frac{a-b x}{2 a}\right )}{a b (1+p) \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0946472, size = 89, normalized size = 1.01 \[ \frac{2^{p-\frac{1}{2}} (b x-a) \left (\frac{b x}{a}+1\right )^{\frac{1}{2}-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (\frac{1}{2}-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1) \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.52, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}{\frac{1}{\sqrt{bx+a}}}}\, 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}}{\sqrt{b x + a}}\,{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}}{\sqrt{b x + a}}, 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}}{\sqrt{a + b x}}\, 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}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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