3.2382 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=194 \[ -\frac{3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac{9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac{24269487 \sqrt{5 x+3} (1-2 x)^{7/2}}{20480000}+\frac{88988119 \sqrt{5 x+3} (1-2 x)^{5/2}}{204800000}+\frac{978869309 \sqrt{5 x+3} (1-2 x)^{3/2}}{819200000}+\frac{32302687197 \sqrt{5 x+3} \sqrt{1-2 x}}{8192000000}+\frac{355329559167 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000000 \sqrt{10}} \]

[Out]

(32302687197*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000000 + (978869309*(1 - 2*x)^(3/2
)*Sqrt[3 + 5*x])/819200000 + (88988119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/204800000
- (24269487*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/20480000 - (735439*(1 - 2*x)^(7/2)*(3
+ 5*x)^(3/2))/1280000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 - (9
*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2)*(18399 + 13480*x))/448000 + (355329559167*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000000*Sqrt[10])

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Rubi [A]  time = 0.241778, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac{9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac{24269487 \sqrt{5 x+3} (1-2 x)^{7/2}}{20480000}+\frac{88988119 \sqrt{5 x+3} (1-2 x)^{5/2}}{204800000}+\frac{978869309 \sqrt{5 x+3} (1-2 x)^{3/2}}{819200000}+\frac{32302687197 \sqrt{5 x+3} \sqrt{1-2 x}}{8192000000}+\frac{355329559167 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(32302687197*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000000 + (978869309*(1 - 2*x)^(3/2
)*Sqrt[3 + 5*x])/819200000 + (88988119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/204800000
- (24269487*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/20480000 - (735439*(1 - 2*x)^(7/2)*(3
+ 5*x)^(3/2))/1280000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 - (9
*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2)*(18399 + 13480*x))/448000 + (355329559167*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000000*Sqrt[10])

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Rubi in Sympy [A]  time = 21.2302, size = 178, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{5}{2}}}{80} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}} \left (90990 x + \frac{496773}{4}\right )}{336000} + \frac{735439 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3200000} + \frac{8089829 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{25600000} + \frac{88988119 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{256000000} - \frac{978869309 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{2048000000} - \frac{32302687197 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8192000000} + \frac{355329559167 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{81920000000} \]

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)**(3/2),x)

[Out]

-3*(-2*x + 1)**(7/2)*(3*x + 2)**2*(5*x + 3)**(5/2)/80 - (-2*x + 1)**(7/2)*(5*x +
3)**(5/2)*(90990*x + 496773/4)/336000 + 735439*(-2*x + 1)**(5/2)*(5*x + 3)**(5/
2)/3200000 + 8089829*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/25600000 + 88988119*sqrt
(-2*x + 1)*(5*x + 3)**(5/2)/256000000 - 978869309*sqrt(-2*x + 1)*(5*x + 3)**(3/2
)/2048000000 - 32302687197*sqrt(-2*x + 1)*sqrt(5*x + 3)/8192000000 + 35532955916
7*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/81920000000

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Mathematica [A]  time = 0.15307, size = 85, normalized size = 0.44 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (3870720000000 x^7+7105536000000 x^6+808627200000 x^5-5264367872000 x^4-2347326614400 x^3+1304824422880 x^2+942468770660 x-115416461871\right )-2487306914169 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{573440000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-115416461871 + 942468770660*x + 1304824422880*
x^2 - 2347326614400*x^3 - 5264367872000*x^4 + 808627200000*x^5 + 7105536000000*x
^6 + 3870720000000*x^7) - 2487306914169*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]
])/573440000000

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Maple [A]  time = 0.014, size = 172, normalized size = 0.9 \[{\frac{1}{1146880000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 77414400000000\,{x}^{7}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-105287357440000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-46946532288000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-2308329237420\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1146880000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(77414400000000*x^7*(-10*x^2-x+3)^(1
/2)[phone]*x^6*(-10*x^2-x+3)^(1/2)[phone]*x^5*(-10*x^2-x+3)^(1/
2)-105287357440000*x^4*(-10*x^2-x+3)^(1/2)-46946532288000*x^3*(-10*x^2-x+3)^(1/2
)[phone]*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)-2308329237420*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48951, size = 180, normalized size = 0.93 \[ \frac{27}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{6183}{5600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{71331}{224000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{6491477}{22400000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8089829}{5120000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{8089829}{102400000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{2936607927}{409600000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{355329559167}{163840000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{2936607927}{8192000000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/40*(-10*x^2 - x + 3)^(5/2)*x^3 + 6183/5600*(-10*x^2 - x + 3)^(5/2)*x^2 + 7133
1/224000*(-10*x^2 - x + 3)^(5/2)*x - 6491477/22400000*(-10*x^2 - x + 3)^(5/2) +
8089829/5120000*(-10*x^2 - x + 3)^(3/2)*x + 8089829/102400000*(-10*x^2 - x + 3)^
(3/2) + 2936607927/409600000*sqrt(-10*x^2 - x + 3)*x - 355329559167/163840000000
*sqrt(10)*arcsin(-20/11*x - 1/11) + 2936607927/8192000000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.225291, size = 117, normalized size = 0.6 \[ \frac{1}{1146880000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (3870720000000 \, x^{7} + 7105536000000 \, x^{6} + 808627200000 \, x^{5} - 5264367872000 \, x^{4} - 2347326614400 \, x^{3} + 1304824422880 \, x^{2} + 942468770660 \, x - 115416461871\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2487306914169 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1146880000000*sqrt(10)*(2*sqrt(10)*(3870720000000*x^7 + 7105536000000*x^6 + 80
8627200000*x^5 - 5264367872000*x^4 - 2347326614400*x^3 + 1304824422880*x^2 + 942
468770660*x - 115416461871)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 2487306914169*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)**3*(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291845, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done