3.2322 \(\int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{407}{640} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{4477 \sqrt{5 x+3} (1-2 x)^{3/2}}{12800}+\frac{147741 \sqrt{5 x+3} \sqrt{1-2 x}}{128000}+\frac{1625151 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

[Out]

(147741*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128000 + (4477*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
])/12800 - (407*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/640 - (37*(1 - 2*x)^(5/2)*(3 + 5*
x)^(3/2))/160 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/50 + (1625151*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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Rubi [A]  time = 0.144122, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{407}{640} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{4477 \sqrt{5 x+3} (1-2 x)^{3/2}}{12800}+\frac{147741 \sqrt{5 x+3} \sqrt{1-2 x}}{128000}+\frac{1625151 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2),x]

[Out]

(147741*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128000 + (4477*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
])/12800 - (407*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/640 - (37*(1 - 2*x)^(5/2)*(3 + 5*
x)^(3/2))/160 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/50 + (1625151*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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Rubi in Sympy [A]  time = 12.532, size = 126, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{50} + \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{400} + \frac{407 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{4000} - \frac{4477 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{32000} - \frac{147741 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{128000} + \frac{1625151 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1280000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(2+3*x)*(3+5*x)**(3/2),x)

[Out]

-3*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/50 + 37*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)
/400 + 407*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/4000 - 4477*sqrt(-2*x + 1)*(5*x + 3)*
*(3/2)/32000 - 147741*sqrt(-2*x + 1)*sqrt(5*x + 3)/128000 + 1625151*sqrt(10)*asi
n(sqrt(22)*sqrt(5*x + 3)/11)/1280000

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Mathematica [A]  time = 0.084432, size = 70, normalized size = 0.51 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (768000 x^4+745600 x^3-364320 x^2-489340 x+46809\right )-1625151 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1280000} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2),x]

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(46809 - 489340*x - 364320*x^2 + 745600*x^3 + 7
68000*x^4) - 1625151*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1280000

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Maple [A]  time = 0.012, size = 121, normalized size = 0.9 \[{\frac{1}{2560000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -15360000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-14912000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7286400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1625151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +9786800\,x\sqrt{-10\,{x}^{2}-x+3}-936180\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^(3/2),x)

[Out]

1/2560000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-15360000*x^4*(-10*x^2-x+3)^(1/2)-1491200
0*x^3*(-10*x^2-x+3)^(1/2)+7286400*x^2*(-10*x^2-x+3)^(1/2)+1625151*10^(1/2)*arcsi
n(20/11*x+1/11)+9786800*x*(-10*x^2-x+3)^(1/2)-936180*(-10*x^2-x+3)^(1/2))/(-10*x
^2-x+3)^(1/2)

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Maxima [A]  time = 1.49318, size = 113, normalized size = 0.82 \[ -\frac{3}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{37}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{37}{1600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{13431}{6400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1625151}{2560000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{13431}{128000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

-3/50*(-10*x^2 - x + 3)^(5/2) + 37/80*(-10*x^2 - x + 3)^(3/2)*x + 37/1600*(-10*x
^2 - x + 3)^(3/2) + 13431/6400*sqrt(-10*x^2 - x + 3)*x - 1625151/2560000*sqrt(10
)*arcsin(-20/11*x - 1/11) + 13431/128000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.219688, size = 97, normalized size = 0.7 \[ -\frac{1}{2560000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (768000 \, x^{4} + 745600 \, x^{3} - 364320 \, x^{2} - 489340 \, x + 46809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1625151 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/2560000*sqrt(10)*(2*sqrt(10)*(768000*x^4 + 745600*x^3 - 364320*x^2 - 489340*x
+ 46809)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1625151*arctan(1/20*sqrt(10)*(20*x + 1)
/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(3/2)*(2+3*x)*(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.244911, size = 317, normalized size = 2.3 \[ -\frac{1}{6400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{23}{1920000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{7}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2),x, algorithm="giac")

[Out]

-1/6400000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13640
5)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) - 23/1920000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) +
2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3))) + 7/24000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*s
qrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
+ 3/200*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin
(1/11*sqrt(22)*sqrt(5*x + 3)))