3.2409 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{3}{50} (3 x+2) \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{369 \sqrt{5 x+3} (1-2 x)^{7/2}}{4000}+\frac{4907 \sqrt{5 x+3} (1-2 x)^{5/2}}{120000}+\frac{53977 \sqrt{5 x+3} (1-2 x)^{3/2}}{480000}+\frac{593747 \sqrt{5 x+3} \sqrt{1-2 x}}{1600000}+\frac{6531217 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600000 \sqrt{10}} \]

[Out]

(593747*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600000 + (53977*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/480000 + (4907*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/120000 - (369*(1 - 2*x)^(7/2)
*Sqrt[3 + 5*x])/4000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 + 5*x])/50 + (6531217
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600000*Sqrt[10])

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Rubi [A]  time = 0.168452, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{50} (3 x+2) \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{369 \sqrt{5 x+3} (1-2 x)^{7/2}}{4000}+\frac{4907 \sqrt{5 x+3} (1-2 x)^{5/2}}{120000}+\frac{53977 \sqrt{5 x+3} (1-2 x)^{3/2}}{480000}+\frac{593747 \sqrt{5 x+3} \sqrt{1-2 x}}{1600000}+\frac{6531217 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(593747*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600000 + (53977*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/480000 + (4907*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/120000 - (369*(1 - 2*x)^(7/2)
*Sqrt[3 + 5*x])/4000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 + 5*x])/50 + (6531217
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600000*Sqrt[10])

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Rubi in Sympy [A]  time = 13.4584, size = 129, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3} \left (9 x + 6\right )}{50} - \frac{369 \left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{4000} + \frac{4907 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{120000} + \frac{53977 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{480000} + \frac{593747 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1600000} + \frac{6531217 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)*sqrt(5*x + 3)*(9*x + 6)/50 - 369*(-2*x + 1)**(7/2)*sqrt(5*x +
3)/4000 + 4907*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/120000 + 53977*(-2*x + 1)**(3/2)
*sqrt(5*x + 3)/480000 + 593747*sqrt(-2*x + 1)*sqrt(5*x + 3)/1600000 + 6531217*sq
rt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/16000000

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Mathematica [A]  time = 0.119292, size = 70, normalized size = 0.49 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4-2217600 x^3-6256480 x^2+3384140 x+1498491\right )-19593651 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{48000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1498491 + 3384140*x - 6256480*x^2 - 2217600*x^3
+ 6912000*x^4) - 19593651*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/48000000

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Maple [A]  time = 0.014, size = 121, normalized size = 0.9 \[{\frac{1}{96000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-44352000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-125129600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)-443520
00*x^3*(-10*x^2-x+3)^(1/2)-125129600*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*a
rcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))
/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50929, size = 124, normalized size = 0.87 \[ \frac{36}{25} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{231}{500} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{39103}{30000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{169207}{240000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6531217}{32000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{499497}{1600000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

36/25*sqrt(-10*x^2 - x + 3)*x^4 - 231/500*sqrt(-10*x^2 - x + 3)*x^3 - 39103/3000
0*sqrt(-10*x^2 - x + 3)*x^2 + 169207/240000*sqrt(-10*x^2 - x + 3)*x - 6531217/32
000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 499497/1600000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222396, size = 97, normalized size = 0.68 \[ \frac{1}{96000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6912000 \, x^{4} - 2217600 \, x^{3} - 6256480 \, x^{2} + 3384140 \, x + 1498491\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19593651 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96000000*sqrt(10)*(2*sqrt(10)*(6912000*x^4 - 2217600*x^3 - 6256480*x^2 + 33841
40*x + 1498491)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 19593651*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)**(5/2)*(2+3*x)**2/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259698, size = 371, normalized size = 2.59 \[ \frac{3}{80000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 203\right )}{\left (5 \, x + 3\right )} + 19073\right )}{\left (5 \, x + 3\right )} - 506185\right )}{\left (5 \, x + 3\right )} + 4031895\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 10392195 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{23}{120000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{2}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 5061
85)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin
(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/800000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3
) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3))) - 23/120000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3)
+ 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(
5*x + 3))) - 1/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sq
rt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))