3.2419 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt{5 x+3}}+\frac{13}{50} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^3+\frac{111 (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2}{5000}-\frac{(1-2 x)^{5/2} \sqrt{5 x+3} (1990620 x+2725981)}{8000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{5 x+3}}{32000000}+\frac{118054167 \sqrt{1-2 x} \sqrt{5 x+3}}{320000000}+\frac{1298595837 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (118054167*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/320000000 + (3577399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/32000000 + (111
*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/5000 + (13*(1 - 2*x)^(5/2)*(2 + 3*x)
^3*Sqrt[3 + 5*x])/50 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(2725981 + 1990620*x))/800
0000 + (1298595837*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320000000*Sqrt[10])

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Rubi [A]  time = 0.314171, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt{5 x+3}}+\frac{13}{50} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^3+\frac{111 (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2}{5000}-\frac{(1-2 x)^{5/2} \sqrt{5 x+3} (1990620 x+2725981)}{8000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{5 x+3}}{32000000}+\frac{118054167 \sqrt{1-2 x} \sqrt{5 x+3}}{320000000}+\frac{1298595837 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^(3/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (118054167*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/320000000 + (3577399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/32000000 + (111
*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/5000 + (13*(1 - 2*x)^(5/2)*(2 + 3*x)
^3*Sqrt[3 + 5*x])/50 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(2725981 + 1990620*x))/800
0000 + (1298595837*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320000000*Sqrt[10])

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Rubi in Sympy [A]  time = 30.9993, size = 173, normalized size = 0.93 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{4}}{5 \sqrt{5 x + 3}} + \frac{13 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{50} + \frac{111 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{5000} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3} \left (\frac{4478895 x}{2} + \frac{24533829}{8}\right )}{9000000} + \frac{3577399 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{32000000} + \frac{118054167 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{320000000} + \frac{1298595837 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3200000000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**4/(5*sqrt(5*x + 3)) + 13*(-2*x + 1)**(5/2)*(3*x
+ 2)**3*sqrt(5*x + 3)/50 + 111*(-2*x + 1)**(5/2)*(3*x + 2)**2*sqrt(5*x + 3)/5000
- (-2*x + 1)**(5/2)*sqrt(5*x + 3)*(4478895*x/2 + 24533829/8)/9000000 + 3577399*
(-2*x + 1)**(3/2)*sqrt(5*x + 3)/32000000 + 118054167*sqrt(-2*x + 1)*sqrt(5*x + 3
)/320000000 + 1298595837*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/3200000000

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Mathematica [A]  time = 0.202332, size = 84, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \left (3456000000 x^6+4043520000 x^5-2530224000 x^4-3673002400 x^3+938891620 x^2+1366129125 x+168414751\right )-1298595837 \sqrt{50 x+30} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3200000000 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^(3/2),x]

[Out]

(10*Sqrt[1 - 2*x]*(168414751 + 1366129125*x + 938891620*x^2 - 3673002400*x^3 - 2
530224000*x^4 + 4043520000*x^5 + 3456000000*x^6) - 1298595837*Sqrt[30 + 50*x]*Ar
cSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3200000000*Sqrt[3 + 5*x])

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Maple [A]  time = 0.02, size = 167, normalized size = 0.9 \[{\frac{1}{6400000000} \left ( 69120000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-50604480000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-73460048000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+18777832400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(3/2),x)

[Out]

1/6400000000*(69120000000*x^6*(-10*x^2-x+3)^(1/2)[phone]*x^5*(-10*x^2-x+3)^
(1/2)-50604480000*x^4*(-10*x^2-x+3)^(1/2)-73460048000*x^3*(-10*x^2-x+3)^(1/2)+64
92979185*10^(1/2)*arcsin(20/11*x+1/11)*x+18777832400*x^2*(-10*x^2-x+3)^(1/2)+389
5787511*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]
20*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.52235, size = 193, normalized size = 1.04 \[ -\frac{108 \, x^{7}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1809 \, x^{6}}{125 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{284499 \, x^{5}}{10000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3009863 \, x^{4}}{200000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{138769641 \, x^{3}}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{179336663 \, x^{2}}{32000000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1298595837}{6400000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1029299623 \, x}{320000000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{168414751}{320000000 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(3/2),x, algorithm="maxima")

[Out]

-108/5*x^7/sqrt(-10*x^2 - x + 3) - 1809/125*x^6/sqrt(-10*x^2 - x + 3) + 284499/1
0000*x^5/sqrt(-10*x^2 - x + 3) + 3009863/200000*x^4/sqrt(-10*x^2 - x + 3) - 1387
69641/8000000*x^3/sqrt(-10*x^2 - x + 3) - 179336663/32000000*x^2/sqrt(-10*x^2 -
x + 3) - 1298595837/6400000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1029299623/320
000000*x/sqrt(-10*x^2 - x + 3) + 168414751/320000000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.228631, size = 127, normalized size = 0.68 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (3456000000 \, x^{6} + 4043520000 \, x^{5} - 2530224000 \, x^{4} - 3673002400 \, x^{3} + 938891620 \, x^{2} + 1366129125 \, x + 168414751\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1298595837 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6400000000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(3/2),x, algorithm="fricas")

[Out]

1/6400000000*sqrt(10)*(2*sqrt(10)*(3456000000*x^6 + 4043520000*x^5 - 2530224000*
x^4 - 3673002400*x^3 + 938891620*x^2 + 1366129125*x + 168414751)*sqrt(5*x + 3)*s
qrt(-2*x + 1) + 1298595837*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))/(5*x + 3)

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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)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.352446, size = 220, normalized size = 1.18 \[ \frac{1}{8000000000} \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (20 \, \sqrt{5}{\left (5 \, x + 3\right )} - 243 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9263 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2532859 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3473645 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 533500275 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1298595837}{3200000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{781250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{390625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(3/2),x, algorithm="giac")

[Out]

1/8000000000*(4*(8*(108*(16*(20*sqrt(5)*(5*x + 3) - 243*sqrt(5))*(5*x + 3) + 926
3*sqrt(5))*(5*x + 3) + 2532859*sqrt(5))*(5*x + 3) + 3473645*sqrt(5))*(5*x + 3) -
533500275*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 1298595837/3200000000*sqrt(1
0)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/781250*sqrt(10)*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) + 242/390625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))