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

Optimal. Leaf size=28 \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0959226, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079 \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.2696, size = 112, normalized size = 4. \[ - a x \left (1 - \frac{b^{2} x^{2}}{a^{2}}\right )^{- p} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )} - \frac{2 a \left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*x*(1 - b**2*x**2/a**2)**(-p)*(a**2 - b**2*x**2)**p*hyper((-p, 1/2), (3/2,), b
**2*x**2/a**2) - 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 = 0.00981708, size = 28, normalized size = 1. \[ -\frac{\left (a^2-b^2 x^2\right )^{p+1}}{2 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 1.3 \[ -{\frac{ \left ( -bx+a \right ) \left ( bx+a \right ) \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{2\,b \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.81148, size = 57, normalized size = 2.04 \[ \frac{{\left (b^{2} x^{2} - a^{2}\right )} e^{\left (p \log \left (b x + a\right ) + p \log \left (-b x + a\right )\right )}}{2 \, b{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*x^2 - a^2)*e^(p*log(b*x + a) + p*log(-b*x + a))/(b*(p + 1))

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Fricas [A]  time = 0.233498, size = 49, normalized size = 1.75 \[ \frac{{\left (b^{2} x^{2} - a^{2}\right )}{\left (-b^{2} x^{2} + a^{2}\right )}^{p}}{2 \,{\left (b p + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.71171, size = 49, normalized size = 1.75 \[ b \left (\begin{cases} \frac{x^{2} \left (a^{2}\right )^{p}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\begin{cases} \frac{\left (a^{2} - b^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a^{2} - b^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*Piecewise((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 [A]  time = 0.225507, size = 35, normalized size = 1.25 \[ -\frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{p + 1}}{2 \, b{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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