3.2522 \(\int \frac{(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]

[Out]

(9738340821*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1638400 + (295101237*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/409600 + (999*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/160 + (13*
Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(5/2))/8 + ((2 + 3*x)^4*(3 + 5*x)^(5/2))/Sqr
t[1 - 2*x] + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7611023 + 3765060*x))/51200 - (1071
21749031*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1638400*Sqrt[10])

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Rubi [A]  time = 0.292871, antiderivative size = 183, 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 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{13}{8} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac{999}{160} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac{295101237 \sqrt{1-2 x} (5 x+3)^{3/2}}{409600}+\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac{9738340821 \sqrt{1-2 x} \sqrt{5 x+3}}{1638400}-\frac{107121749031 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1638400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(9738340821*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1638400 + (295101237*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/409600 + (999*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/160 + (13*
Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(5/2))/8 + ((2 + 3*x)^4*(3 + 5*x)^(5/2))/Sqr
t[1 - 2*x] + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7611023 + 3765060*x))/51200 - (1071
21749031*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1638400*Sqrt[10])

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Rubi in Sympy [A]  time = 30.4489, size = 170, normalized size = 0.93 \[ \frac{13 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{5}{2}}}{8} + \frac{999 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{5}{2}}}{160} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}} \left (\frac{1058923125 x}{4} + \frac{8562400875}{16}\right )}{3600000} + \frac{295101237 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{409600} + \frac{9738340821 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1638400} - \frac{107121749031 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16384000} + \frac{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13*sqrt(-2*x + 1)*(3*x + 2)**3*(5*x + 3)**(5/2)/8 + 999*sqrt(-2*x + 1)*(3*x + 2)
**2*(5*x + 3)**(5/2)/160 + sqrt(-2*x + 1)*(5*x + 3)**(5/2)*(1058923125*x/4 + 856
2400875/16)/3600000 + 295101237*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/409600 + 9738340
821*sqrt(-2*x + 1)*sqrt(5*x + 3)/1638400 - 107121749031*sqrt(10)*asin(sqrt(22)*s
qrt(5*x + 3)/11)/16384000 + (3*x + 2)**4*(5*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.13574, size = 84, normalized size = 0.46 \[ \frac{107121749031 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (276480000 x^6+1479168000 x^5+3687379200 x^4+5945485120 x^3+7755469800 x^2+11734056318 x-16267424049\right )}{16384000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-16267424049 + 11734056318*x + 7755469800*x^2 + 5945485120*x
^3 + 3687379200*x^4 + 1479168000*x^5 + 276480000*x^6) + 107121749031*Sqrt[10 - 2
0*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(16384000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.019, size = 174, normalized size = 1. \[ -{\frac{1}{-32768000+65536000\,x} \left ( -5529600000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-29583360000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-73747584000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-118909702400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-155109396000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-107121749031\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -234681126360\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32768000*(-5529600000*x^6*(-10*x^2-x+3)^(1/2)-29583360000*x^5*(-10*x^2-x+3)^(
1/2)-73747584000*x^4*(-10*x^2-x+3)^(1/2)-118909702400*x^3*(-10*x^2-x+3)^(1/2)+21
4243498062*10^(1/2)*arcsin(20/11*x+1/11)*x-155109396000*x^2*(-10*x^2-x+3)^(1/2)-
107121749031*10^(1/2)*arcsin(20/11*x+1/11)-234681126360*x*(-10*x^2-x+3)^(1/2)+32
5348480980*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+
3)^(1/2)

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Maxima [A]  time = 1.51471, size = 193, normalized size = 1.05 \[ -\frac{3375 \, x^{7}}{4 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{80325 \, x^{6}}{16 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3574125 \, x^{5}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{25493477 \, x^{4}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1415345109 \, x^{3}}{40960 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{8193669099 \, x^{2}}{163840 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{107121749031}{32768000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{46134951291 \, x}{1638400 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{48802272147}{1638400 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3375/4*x^7/sqrt(-10*x^2 - x + 3) - 80325/16*x^6/sqrt(-10*x^2 - x + 3) - 3574125
/256*x^5/sqrt(-10*x^2 - x + 3) - 25493477/1024*x^4/sqrt(-10*x^2 - x + 3) - 14153
45109/40960*x^3/sqrt(-10*x^2 - x + 3) - 8193669099/163840*x^2/sqrt(-10*x^2 - x +
3) + 107121749031/32768000*sqrt(10)*arcsin(-20/11*x - 1/11) + 46134951291/16384
00*x/sqrt(-10*x^2 - x + 3) + 48802272147/1638400/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.245612, size = 127, normalized size = 0.69 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 107121749031 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{32768000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32768000*sqrt(10)*(2*sqrt(10)*(276480000*x^6 + 1479168000*x^5 + 3687379200*x^4
+ 5945485120*x^3 + 7755469800*x^2 + 11734056318*x - 16267424049)*sqrt(5*x + 3)*
sqrt(-2*x + 1) - 107121749031*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*
x + 3)*sqrt(-2*x + 1))))/(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((2+3*x)**4*(3+5*x)**(5/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.248553, size = 166, normalized size = 0.91 \[ -\frac{107121749031}{16384000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 35 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4299 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3832457 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 295101237 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16230568035 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 535608745155 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{204800000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107121749031/16384000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/20480000
0*(2*(4*(8*(108*(16*(4*sqrt(5)*(5*x + 3) + 35*sqrt(5))*(5*x + 3) + 4299*sqrt(5))
*(5*x + 3) + 3832457*sqrt(5))*(5*x + 3) + 295101237*sqrt(5))*(5*x + 3) + 1623056
8035*sqrt(5))*(5*x + 3) - 535608745155*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2
*x - 1)