3.7 \(\int \left (3 x-4 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=79 \[ -\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{3645 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{131072} \]

[Out]

(-405*(3 - 8*x)*Sqrt[3*x - 4*x^2])/32768 - (15*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/10
24 - ((3 - 8*x)*(3*x - 4*x^2)^(5/2))/48 - (3645*ArcSin[1 - (8*x)/3])/131072

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Rubi [A]  time = 0.0469482, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac{15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac{405 (3-8 x) \sqrt{3 x-4 x^2}}{32768}-\frac{3645 \sin ^{-1}\left (1-\frac{8 x}{3}\right )}{131072} \]

Antiderivative was successfully verified.

[In]  Int[(3*x - 4*x^2)^(5/2),x]

[Out]

(-405*(3 - 8*x)*Sqrt[3*x - 4*x^2])/32768 - (15*(3 - 8*x)*(3*x - 4*x^2)^(3/2))/10
24 - ((3 - 8*x)*(3*x - 4*x^2)^(5/2))/48 - (3645*ArcSin[1 - (8*x)/3])/131072

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Rubi in Sympy [A]  time = 2.81776, size = 70, normalized size = 0.89 \[ - \frac{\left (- 8 x + 3\right ) \left (- 4 x^{2} + 3 x\right )^{\frac{5}{2}}}{48} - \frac{15 \left (- 8 x + 3\right ) \left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}}{1024} - \frac{405 \left (- 8 x + 3\right ) \sqrt{- 4 x^{2} + 3 x}}{32768} + \frac{3645 \operatorname{asin}{\left (\frac{8 x}{3} - 1 \right )}}{131072} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-4*x**2+3*x)**(5/2),x)

[Out]

-(-8*x + 3)*(-4*x**2 + 3*x)**(5/2)/48 - 15*(-8*x + 3)*(-4*x**2 + 3*x)**(3/2)/102
4 - 405*(-8*x + 3)*sqrt(-4*x**2 + 3*x)/32768 + 3645*asin(8*x/3 - 1)/131072

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Mathematica [A]  time = 0.0610432, size = 92, normalized size = 1.16 \[ \frac{\sqrt{-x (4 x-3)} \left (2 \sqrt{x} \sqrt{4 x-3} \left (262144 x^5-491520 x^4+248832 x^3-3456 x^2-3240 x-3645\right )-10935 \log \left (2 \sqrt{x}+\sqrt{4 x-3}\right )\right )}{196608 \sqrt{x} \sqrt{4 x-3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3*x - 4*x^2)^(5/2),x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(2*Sqrt[x]*Sqrt[-3 + 4*x]*(-3645 - 3240*x - 3456*x^2 + 24
8832*x^3 - 491520*x^4 + 262144*x^5) - 10935*Log[2*Sqrt[x] + Sqrt[-3 + 4*x]]))/(1
96608*Sqrt[x]*Sqrt[-3 + 4*x])

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Maple [A]  time = 0.006, size = 64, normalized size = 0.8 \[ -{\frac{45-120\,x}{1024} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3-8\,x}{48} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{5}{2}}}}+{\frac{3645}{131072}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{1215-3240\,x}{32768}\sqrt{-4\,{x}^{2}+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2+3*x)^(5/2),x)

[Out]

-15/1024*(3-8*x)*(-4*x^2+3*x)^(3/2)-1/48*(3-8*x)*(-4*x^2+3*x)^(5/2)+3645/131072*
arcsin(-1+8/3*x)-405/32768*(3-8*x)*(-4*x^2+3*x)^(1/2)

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Maxima [A]  time = 0.797944, size = 122, normalized size = 1.54 \[ \frac{1}{6} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}} x - \frac{1}{16} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{5}{2}} + \frac{15}{128} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} x - \frac{45}{1024} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{405}{4096} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{1215}{32768} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{3645}{131072} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(5/2),x, algorithm="maxima")

[Out]

1/6*(-4*x^2 + 3*x)^(5/2)*x - 1/16*(-4*x^2 + 3*x)^(5/2) + 15/128*(-4*x^2 + 3*x)^(
3/2)*x - 45/1024*(-4*x^2 + 3*x)^(3/2) + 405/4096*sqrt(-4*x^2 + 3*x)*x - 1215/327
68*sqrt(-4*x^2 + 3*x) - 3645/131072*arcsin(-8/3*x + 1)

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Fricas [A]  time = 0.216161, size = 78, normalized size = 0.99 \[ \frac{1}{98304} \,{\left (262144 \, x^{5} - 491520 \, x^{4} + 248832 \, x^{3} - 3456 \, x^{2} - 3240 \, x - 3645\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{3645}{65536} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(5/2),x, algorithm="fricas")

[Out]

1/98304*(262144*x^5 - 491520*x^4 + 248832*x^3 - 3456*x^2 - 3240*x - 3645)*sqrt(-
4*x^2 + 3*x) - 3645/65536*arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- 4 x^{2} + 3 x\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x**2+3*x)**(5/2),x)

[Out]

Integral((-4*x**2 + 3*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.213834, size = 63, normalized size = 0.8 \[ \frac{1}{98304} \,{\left (8 \,{\left (16 \,{\left (8 \,{\left (32 \,{\left (8 \, x - 15\right )} x + 243\right )} x - 27\right )} x - 405\right )} x - 3645\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{3645}{131072} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x^2 + 3*x)^(5/2),x, algorithm="giac")

[Out]

1/98304*(8*(16*(8*(32*(8*x - 15)*x + 243)*x - 27)*x - 405)*x - 3645)*sqrt(-4*x^2
+ 3*x) + 3645/131072*arcsin(8/3*x - 1)