3.2408 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{\sqrt{5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac{1}{20} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{276493 \sqrt{5 x+3} (1-2 x)^{5/2}}{4800000}+\frac{3041423 \sqrt{5 x+3} (1-2 x)^{3/2}}{19200000}+\frac{33455653 \sqrt{5 x+3} \sqrt{1-2 x}}{64000000}+\frac{368012183 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000000 \sqrt{10}} \]

[Out]

(33455653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000000 + (3041423*(1 - 2*x)^(3/2)*Sqrt[
3 + 5*x])/19200000 + (276493*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4800000 - ((1 - 2*x)
^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/20 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]*(52951 + 4
7280*x))/160000 + (368012183*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000000*Sqrt[10
])

_______________________________________________________________________________________

Rubi [A]  time = 0.191819, antiderivative size = 150, 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 \[ -\frac{\sqrt{5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac{1}{20} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{276493 \sqrt{5 x+3} (1-2 x)^{5/2}}{4800000}+\frac{3041423 \sqrt{5 x+3} (1-2 x)^{3/2}}{19200000}+\frac{33455653 \sqrt{5 x+3} \sqrt{1-2 x}}{64000000}+\frac{368012183 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(33455653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000000 + (3041423*(1 - 2*x)^(3/2)*Sqrt[
3 + 5*x])/19200000 + (276493*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4800000 - ((1 - 2*x)
^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/20 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]*(52951 + 4
7280*x))/160000 + (368012183*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000000*Sqrt[10
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.092, size = 136, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{20} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3} \left (35460 x + \frac{158853}{4}\right )}{120000} + \frac{276493 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{4800000} + \frac{3041423 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{19200000} + \frac{33455653 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{64000000} + \frac{368012183 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{640000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)*(3*x + 2)**2*sqrt(5*x + 3)/20 - (-2*x + 1)**(7/2)*sqrt(5*x +
3)*(35460*x + 158853/4)/120000 + 276493*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/4800000
+ 3041423*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/19200000 + 33455653*sqrt(-2*x + 1)*sqr
t(5*x + 3)/64000000 + 368012183*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/6400000
00

_______________________________________________________________________________________

Mathematica [A]  time = 0.126567, size = 75, normalized size = 0.5 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (691200000 x^5+338688000 x^4-729302400 x^3-233839520 x^2+334643860 x+39899709\right )-1104036549 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1920000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39899709 + 334643860*x - 233839520*x^2 - 729302
400*x^3 + 338688000*x^4 + 691200000*x^5) - 1104036549*Sqrt[10]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/1920000000

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 138, normalized size = 0.9 \[{\frac{1}{3840000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-14586048000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-4676790400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)+67
73760000*x^4*(-10*x^2-x+3)^(1/2)-14586048000*x^3*(-10*x^2-x+3)^(1/2)-4676790400*
x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(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)

_______________________________________________________________________________________

Maxima [A]  time = 1.51633, size = 147, normalized size = 0.98 \[ \frac{18}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + \frac{441}{250} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{75969}{20000} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{1461497}{1200000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{16732193}{9600000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{368012183}{1280000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{13299903}{64000000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/5*sqrt(-10*x^2 - x + 3)*x^5 + 441/250*sqrt(-10*x^2 - x + 3)*x^4 - 75969/20000
*sqrt(-10*x^2 - x + 3)*x^3 - 1461497/1200000*sqrt(-10*x^2 - x + 3)*x^2 + 1673219
3/9600000*sqrt(-10*x^2 - x + 3)*x - 368012183/1280000000*sqrt(10)*arcsin(-20/11*
x - 1/11) + 13299903/64000000*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.220319, size = 104, normalized size = 0.69 \[ \frac{1}{3840000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (691200000 \, x^{5} + 338688000 \, x^{4} - 729302400 \, x^{3} - 233839520 \, x^{2} + 334643860 \, x + 39899709\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1104036549 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840000000*sqrt(10)*(2*sqrt(10)*(691200000*x^5 + 338688000*x^4 - 729302400*x^3
- 233839520*x^2 + 334643860*x + 39899709)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 110403
6549*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

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)**3/(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273386, size = 481, normalized size = 3.21 \[ \frac{9}{3200000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 311\right )}{\left (5 \, x + 3\right )} + 46071\right )}{\left (5 \, x + 3\right )} - 775911\right )}{\left (5 \, x + 3\right )} + 15385695\right )}{\left (5 \, x + 3\right )} - 99422145\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 220189365 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{80000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 203\right )}{\left (5 \, x + 3\right )} + 19073\right )}{\left (5 \, x + 3\right )} - 506185\right )}{\left (5 \, x + 3\right )} + 4031895\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 10392195 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{3}{640000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{29}{60000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{4}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/3200000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3)
- 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/80000000*sqrt(5
)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4
031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3))) - 3/640000*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)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3))) - 29/60000*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*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))