3.2512 \(\int \frac{(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=161 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac{10377 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac{1018114917 \sqrt{1-2 x} \sqrt{5 x+3}}{1024000}-\frac{11199264087 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024000 \sqrt{10}} \]

[Out]

(1018114917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1024000 + (10377*Sqrt[1 - 2*x]*(2 + 3*x
)^2*(3 + 5*x)^(3/2))/1600 + (33*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 +
((2 + 3*x)^4*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(
4772357 + 2253560*x))/256000 - (11199264087*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1
024000*Sqrt[10])

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Rubi [A]  time = 0.258423, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac{10377 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac{1018114917 \sqrt{1-2 x} \sqrt{5 x+3}}{1024000}-\frac{11199264087 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1018114917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1024000 + (10377*Sqrt[1 - 2*x]*(2 + 3*x
)^2*(3 + 5*x)^(3/2))/1600 + (33*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 +
((2 + 3*x)^4*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(
4772357 + 2253560*x))/256000 - (11199264087*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1
024000*Sqrt[10])

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Rubi in Sympy [A]  time = 28.0872, size = 150, normalized size = 0.93 \[ \frac{33 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{20} + \frac{10377 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{1600} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (\frac{190144125 x}{2} + \frac{3221340975}{16}\right )}{1200000} + \frac{1018114917 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1024000} - \frac{11199264087 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{10240000} + \frac{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33*sqrt(-2*x + 1)*(3*x + 2)**3*(5*x + 3)**(3/2)/20 + 10377*sqrt(-2*x + 1)*(3*x +
2)**2*(5*x + 3)**(3/2)/1600 + sqrt(-2*x + 1)*(5*x + 3)**(3/2)*(190144125*x/2 +
3221340975/16)/1200000 + 1018114917*sqrt(-2*x + 1)*sqrt(5*x + 3)/1024000 - 11199
264087*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/10240000 + (3*x + 2)**4*(5*x + 3
)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.122095, size = 79, normalized size = 0.49 \[ \frac{11199264087 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (41472000 x^5+200966400 x^4+461171520 x^3+732415080 x^2+1206337246 x-1702927233\right )}{10240000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-1702927233 + 1206337246*x + 732415080*x^2 + 461171520*x^3 +
200966400*x^4 + 41472000*x^5) + 11199264087*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*S
qrt[1 - 2*x]])/(10240000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.02, size = 157, normalized size = 1. \[ -{\frac{1}{-20480000+40960000\,x} \left ( -829440000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-4019328000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-9223430400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-14648301600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-11199264087\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -24126744920\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20480000*(-829440000*x^5*(-10*x^2-x+3)^(1/2)-4019328000*x^4*(-10*x^2-x+3)^(1/
2)-9223430400*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*
x-14648301600*x^2*(-10*x^2-x+3)^(1/2)-11199264087*10^(1/2)*arcsin(20/11*x+1/11)-
24126744920*x*(-10*x^2-x+3)^(1/2)[phone]*(-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 = 1.52421, size = 267, normalized size = 1.66 \[ \frac{81}{400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} - \frac{6669}{640} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{12607994487}{20480000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1760913}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) - \frac{359469}{12800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14553}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x - \frac{2420847}{51200} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{305613}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} + \frac{540891153}{1024000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (2 \, x - 1\right )}} - \frac{79233 \, \sqrt{-10 \, x^{2} - x + 3}}{64 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/400*(-10*x^2 - x + 3)^(5/2) - 6669/640*(-10*x^2 - x + 3)^(3/2)*x - 1260799448
7/20480000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1760913/25600*I*sqrt(5)*sqrt
(2)*arcsin(20/11*x - 21/11) - 359469/12800*(-10*x^2 - x + 3)^(3/2) + 14553/64*sq
rt(10*x^2 - 21*x + 8)*x - 2420847/51200*sqrt(-10*x^2 - x + 3)*x - 305613/1280*sq
rt(10*x^2 - 21*x + 8) + 540891153/1024000*sqrt(-10*x^2 - x + 3) - 2401/32*(-10*x
^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 1029/16*(-10*x^2 - x + 3)^(3/2)/(2*x - 1)
- 79233/64*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.229568, size = 120, normalized size = 0.75 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (41472000 \, x^{5} + 200966400 \, x^{4} + 461171520 \, x^{3} + 732415080 \, x^{2} + 1206337246 \, x - 1702927233\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 11199264087 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{20480000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20480000*sqrt(10)*(2*sqrt(10)*(41472000*x^5 + 200966400*x^4 + 461171520*x^3 +
732415080*x^2 + 1206337246*x - 1702927233)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 111992
64087*(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)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247335, size = 149, normalized size = 0.93 \[ -\frac{11199264087}{10240000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 443 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 44497 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 10283927 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1696858195 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 55996320435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{128000000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11199264087/10240000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/128000000
*(2*(12*(24*(12*(48*sqrt(5)*(5*x + 3) + 443*sqrt(5))*(5*x + 3) + 44497*sqrt(5))*
(5*x + 3) + 10283927*sqrt(5))*(5*x + 3) + 1696858195*sqrt(5))*(5*x + 3) - 559963
20435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)