3.958 \(\int (a+b x) \left (a^2-b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=83 \[ a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )-\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]

[Out]

-(a^2 - b^2*x^2)^(1 + p)/(2*b*(1 + p)) + (a*x*(a^2 - b^2*x^2)^p*Hypergeometric2F
1[1/2, -p, 3/2, (b^2*x^2)/a^2])/(1 - (b^2*x^2)/a^2)^p

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Rubi [A]  time = 0.069836, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ a x \left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )-\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(a^2 - b^2*x^2)^p,x]

[Out]

-(a^2 - b^2*x^2)^(1 + p)/(2*b*(1 + p)) + (a*x*(a^2 - b^2*x^2)^p*Hypergeometric2F
1[1/2, -p, 3/2, (b^2*x^2)/a^2])/(1 - (b^2*x^2)/a^2)^p

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Rubi in Sympy [A]  time = 24.7491, size = 66, normalized size = 0.8 \[ - \frac{2 a \left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b**2*x**2+a**2)**p,x)

[Out]

-2*a*((a/2 + b*x/2)/a)**(-p)*(a - b*x)**(-p)*(a - b*x)**(p + 1)*(a**2 - b**2*x**
2)**p*hyper((-p - 1, p + 1), (p + 2,), (a/2 - b*x/2)/a)/(b*(p + 1))

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Mathematica [A]  time = 0.0794374, size = 116, normalized size = 1.4 \[ \frac{\left (a^2-b^2 x^2\right )^p \left (1-\frac{b^2 x^2}{a^2}\right )^{-p} \left (2 a b (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b^2 x^2}{a^2}\right )+b^2 x^2 \left (1-\frac{b^2 x^2}{a^2}\right )^p-a^2 \left (\left (1-\frac{b^2 x^2}{a^2}\right )^p-1\right )\right )}{2 b (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(a^2 - b^2*x^2)^p,x]

[Out]

((a^2 - b^2*x^2)^p*(b^2*x^2*(1 - (b^2*x^2)/a^2)^p - a^2*(-1 + (1 - (b^2*x^2)/a^2
)^p) + 2*a*b*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (b^2*x^2)/a^2]))/(2*b*(1
+ p)*(1 - (b^2*x^2)/a^2)^p)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b^2*x^2+a^2)^p,x)

[Out]

int((b*x+a)*(-b^2*x^2+a^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(-b^2*x^2 + a^2)^p,x, algorithm="maxima")

[Out]

integrate((b*x + a)*(-b^2*x^2 + a^2)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(-b^2*x^2 + a^2)^p,x, algorithm="fricas")

[Out]

integral((b*x + a)*(-b^2*x^2 + a^2)^p, x)

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Sympy [A]  time = 6.199, size = 82, normalized size = 0.99 \[ a a^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )} + b \left (\begin{cases} \frac{x^{2} \left (a^{2}\right )^{p}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\begin{cases} \frac{\left (a^{2} - b^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a^{2} - b^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b**2*x**2+a**2)**p,x)

[Out]

a*a**(2*p)*x*hyper((1/2, -p), (3/2,), b**2*x**2*exp_polar(2*I*pi)/a**2) + b*Piec
ewise((x**2*(a**2)**p/2, Eq(b**2, 0)), (-Piecewise(((a**2 - b**2*x**2)**(p + 1)/
(p + 1), Ne(p, -1)), (log(a**2 - b**2*x**2), True))/(2*b**2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(-b^2*x^2 + a^2)^p,x, algorithm="giac")

[Out]

integrate((b*x + a)*(-b^2*x^2 + a^2)^p, x)