Optimal. Leaf size=88 \[ -\frac{2^{p+\frac{1}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0587131, 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} \[ -\frac{2^{p+\frac{1}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 680
Rule 678
Rule 69
Rubi steps
\begin{align*} \int \sqrt{a+b x} \left (a^2-b^2 x^2\right )^p \, dx &=\frac{\sqrt{a+b x} \int \sqrt{1+\frac{b x}{a}} \left (a^2-b^2 x^2\right )^p \, dx}{\sqrt{1+\frac{b x}{a}}}\\ &=\left (\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{1}{2}+p} \left (a^2-a b x\right )^p \, dx\\ &=-\frac{2^{\frac{1}{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{1}{2}-p,1+p;2+p;\frac{a-b x}{2 a}\right )}{a b (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0936701, size = 89, normalized size = 1.01 \[ \frac{2^{p+\frac{1}{2}} (b x-a) \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.516, size = 0, normalized size = 0. \begin{align*} \int \sqrt{bx+a} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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} \sqrt{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x + a}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]