3.2373 \(\int (1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=138 \[ -\frac{3}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1309 \sqrt{5 x+3} (1-2 x)^{5/2}}{24000}+\frac{14399 \sqrt{5 x+3} (1-2 x)^{3/2}}{96000}+\frac{158389 \sqrt{5 x+3} \sqrt{1-2 x}}{320000}+\frac{1742279 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000 \sqrt{10}} \]

[Out]

(158389*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320000 + (14399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/96000 + (1309*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24000 - (119*(1 - 2*x)^(7/2)*Sq
rt[3 + 5*x])/800 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/50 + (1742279*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/(320000*Sqrt[10])

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Rubi [A]  time = 0.141159, 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 \[ -\frac{3}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1309 \sqrt{5 x+3} (1-2 x)^{5/2}}{24000}+\frac{14399 \sqrt{5 x+3} (1-2 x)^{3/2}}{96000}+\frac{158389 \sqrt{5 x+3} \sqrt{1-2 x}}{320000}+\frac{1742279 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(158389*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320000 + (14399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/96000 + (1309*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24000 - (119*(1 - 2*x)^(7/2)*Sq
rt[3 + 5*x])/800 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/50 + (1742279*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/(320000*Sqrt[10])

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Rubi in Sympy [A]  time = 12.1929, size = 126, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{50} + \frac{119 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{2000} + \frac{1309 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{12000} - \frac{14399 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{32000} + \frac{158389 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{320000} + \frac{1742279 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3200000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)*(5*x + 3)**(3/2)/50 + 119*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2
)/2000 + 1309*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/12000 - 14399*(-2*x + 1)**(3/2)
*sqrt(5*x + 3)/32000 + 158389*sqrt(-2*x + 1)*sqrt(5*x + 3)/320000 + 1742279*sqrt
(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/3200000

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Mathematica [A]  time = 0.0803001, size = 70, normalized size = 0.51 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (2304000 x^4-931200 x^3-1849760 x^2+1108180 x+355917\right )-5226837 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{9600000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(355917 + 1108180*x - 1849760*x^2 - 931200*x^3 +
2304000*x^4) - 5226837*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/9600000

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Maple [A]  time = 0.013, size = 121, normalized size = 0.9 \[{\frac{1}{19200000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 46080000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-18624000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36995200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+5226837\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+7118340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19200000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(46080000*x^4*(-10*x^2-x+3)^(1/2)-1862400
0*x^3*(-10*x^2-x+3)^(1/2)-36995200*x^2*(-10*x^2-x+3)^(1/2)+5226837*10^(1/2)*arcs
in(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)+7118340*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48724, size = 117, normalized size = 0.85 \[ -\frac{6}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{121}{1000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1303}{12000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14399}{16000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1742279}{6400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{14399}{320000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 121/1000*(-10*x^2 - x + 3)^(3/2)*x + 1303/12
000*(-10*x^2 - x + 3)^(3/2) + 14399/16000*sqrt(-10*x^2 - x + 3)*x - 1742279/6400
000*sqrt(10)*arcsin(-20/11*x - 1/11) + 14399/320000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.216821, size = 97, normalized size = 0.7 \[ \frac{1}{19200000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (2304000 \, x^{4} - 931200 \, x^{3} - 1849760 \, x^{2} + 1108180 \, x + 355917\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 5226837 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19200000*sqrt(10)*(2*sqrt(10)*(2304000*x^4 - 931200*x^3 - 1849760*x^2 + 110818
0*x + 355917)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 5226837*arctan(1/20*sqrt(10)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 118.586, size = 490, normalized size = 3.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

242*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*
x + 3)/121 + asin(sqrt(22)*sqrt(5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/3125
+ 638*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqr
t(5*x + 3)/1936 - sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + asin(sqrt(2
2)*sqrt(5*x + 3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/3125 - 256*sqrt(5)*Piecewi
se((14641*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 - sqr
t(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x
+ 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(
sqrt(22)*sqrt(5*x + 3)/11)/128)/16, (x >= -3/5) & (x < 1/2)))/3125 + 24*sqrt(5)*
Piecewise((161051*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/77
44 + 2*sqrt(2)*(-10*x + 5)**(5/2)*(5*x + 3)**(5/2)/805255 - sqrt(2)*(-10*x + 5)*
*(3/2)*(5*x + 3)**(3/2)/3993 - 3*sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)*(-12100*x
- 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/3748096 + 7*asin(sqrt(22)*sqrt(5
*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/3125

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GIAC/XCAS [A]  time = 0.253745, size = 317, normalized size = 2.3 \[ \frac{1}{16000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{480000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{4800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16000000*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))) - 1/480000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 21
79)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) - 1/4800*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt
(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1
/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)))