3.2357 \(\int (7-2 x) \sqrt{9+16 x-4 x^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac{3}{2} (2-x) \sqrt{-4 x^2+16 x+9}-\frac{75}{4} \sin ^{-1}\left (\frac{2 (2-x)}{5}\right ) \]

[Out]

(-3*(2 - x)*Sqrt[9 + 16*x - 4*x^2])/2 + (9 + 16*x - 4*x^2)^(3/2)/6 - (75*ArcSin[
(2*(2 - x))/5])/4

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Rubi [A]  time = 0.0546643, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac{3}{2} (2-x) \sqrt{-4 x^2+16 x+9}-\frac{75}{4} \sin ^{-1}\left (\frac{2 (2-x)}{5}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(7 - 2*x)*Sqrt[9 + 16*x - 4*x^2],x]

[Out]

(-3*(2 - x)*Sqrt[9 + 16*x - 4*x^2])/2 + (9 + 16*x - 4*x^2)^(3/2)/6 - (75*ArcSin[
(2*(2 - x))/5])/4

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Rubi in Sympy [A]  time = 5.95579, size = 61, normalized size = 1.09 \[ - \frac{3 \left (- 8 x + 16\right ) \sqrt{- 4 x^{2} + 16 x + 9}}{16} + \frac{\left (- 4 x^{2} + 16 x + 9\right )^{\frac{3}{2}}}{6} - \frac{75 \operatorname{atan}{\left (\frac{- 8 x + 16}{4 \sqrt{- 4 x^{2} + 16 x + 9}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((7-2*x)*(-4*x**2+16*x+9)**(1/2),x)

[Out]

-3*(-8*x + 16)*sqrt(-4*x**2 + 16*x + 9)/16 + (-4*x**2 + 16*x + 9)**(3/2)/6 - 75*
atan((-8*x + 16)/(4*sqrt(-4*x**2 + 16*x + 9)))/4

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Mathematica [A]  time = 0.0370512, size = 41, normalized size = 0.73 \[ \frac{75}{4} \sin ^{-1}\left (\frac{2 (x-2)}{5}\right )-\frac{1}{6} \sqrt{-4 x^2+16 x+9} \left (4 x^2-25 x+9\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(7 - 2*x)*Sqrt[9 + 16*x - 4*x^2],x]

[Out]

-(Sqrt[9 + 16*x - 4*x^2]*(9 - 25*x + 4*x^2))/6 + (75*ArcSin[(2*(-2 + x))/5])/4

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Maple [A]  time = 0.011, size = 43, normalized size = 0.8 \[ -{\frac{-24\,x+48}{16}\sqrt{-4\,{x}^{2}+16\,x+9}}+{\frac{75}{4}\arcsin \left ( -{\frac{4}{5}}+{\frac{2\,x}{5}} \right ) }+{\frac{1}{6} \left ( -4\,{x}^{2}+16\,x+9 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((7-2*x)*(-4*x^2+16*x+9)^(1/2),x)

[Out]

-3/16*(-8*x+16)*(-4*x^2+16*x+9)^(1/2)+75/4*arcsin(-4/5+2/5*x)+1/6*(-4*x^2+16*x+9
)^(3/2)

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Maxima [A]  time = 0.751502, size = 70, normalized size = 1.25 \[ \frac{1}{6} \,{\left (-4 \, x^{2} + 16 \, x + 9\right )}^{\frac{3}{2}} + \frac{3}{2} \, \sqrt{-4 \, x^{2} + 16 \, x + 9} x - 3 \, \sqrt{-4 \, x^{2} + 16 \, x + 9} - \frac{75}{4} \, \arcsin \left (-\frac{2}{5} \, x + \frac{4}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-4*x^2 + 16*x + 9)*(2*x - 7),x, algorithm="maxima")

[Out]

1/6*(-4*x^2 + 16*x + 9)^(3/2) + 3/2*sqrt(-4*x^2 + 16*x + 9)*x - 3*sqrt(-4*x^2 +
16*x + 9) - 75/4*arcsin(-2/5*x + 4/5)

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Fricas [A]  time = 0.228319, size = 231, normalized size = 4.12 \[ \frac{1872 \, x^{6} - 12276 \, x^{5} - 20160 \, x^{4} + 140049 \, x^{3} + 151146 \, x^{2} - 225 \,{\left (88 \, x^{3} - 1053 \, x^{2} + 9 \,{\left (13 \, x^{2} + 48 \, x + 27\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} - 1944 \, x - 729\right )} \arctan \left (\frac{\sqrt{-4 \, x^{2} + 16 \, x + 9} - 3}{2 \, x}\right ) -{\left (352 \, x^{5} - 6412 \, x^{4} + 19341 \, x^{3} + 39366 \, x^{2} + 12393 \, x\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} + 37179 \, x}{6 \,{\left (88 \, x^{3} - 1053 \, x^{2} + 9 \,{\left (13 \, x^{2} + 48 \, x + 27\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} - 1944 \, x - 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-4*x^2 + 16*x + 9)*(2*x - 7),x, algorithm="fricas")

[Out]

1/6*(1872*x^6 - 12276*x^5 - 20160*x^4 + 140049*x^3 + 151146*x^2 - 225*(88*x^3 -
1053*x^2 + 9*(13*x^2 + 48*x + 27)*sqrt(-4*x^2 + 16*x + 9) - 1944*x - 729)*arctan
(1/2*(sqrt(-4*x^2 + 16*x + 9) - 3)/x) - (352*x^5 - 6412*x^4 + 19341*x^3 + 39366*
x^2 + 12393*x)*sqrt(-4*x^2 + 16*x + 9) + 37179*x)/(88*x^3 - 1053*x^2 + 9*(13*x^2
+ 48*x + 27)*sqrt(-4*x^2 + 16*x + 9) - 1944*x - 729)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int 2 x \sqrt{- 4 x^{2} + 16 x + 9}\, dx - \int \left (- 7 \sqrt{- 4 x^{2} + 16 x + 9}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7-2*x)*(-4*x**2+16*x+9)**(1/2),x)

[Out]

-Integral(2*x*sqrt(-4*x**2 + 16*x + 9), x) - Integral(-7*sqrt(-4*x**2 + 16*x + 9
), x)

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GIAC/XCAS [A]  time = 0.211665, size = 43, normalized size = 0.77 \[ -\frac{1}{6} \,{\left ({\left (4 \, x - 25\right )} x + 9\right )} \sqrt{-4 \, x^{2} + 16 \, x + 9} + \frac{75}{4} \, \arcsin \left (\frac{2}{5} \, x - \frac{4}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(-4*x^2 + 16*x + 9)*(2*x - 7),x, algorithm="giac")

[Out]

-1/6*((4*x - 25)*x + 9)*sqrt(-4*x^2 + 16*x + 9) + 75/4*arcsin(2/5*x - 4/5)