3.2394 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=216 \[ -\frac{1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac{5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac{(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac{63658463 \sqrt{5 x+3} (1-2 x)^{7/2}}{16384000}+\frac{700243093 \sqrt{5 x+3} (1-2 x)^{5/2}}{491520000}+\frac{7702674023 \sqrt{5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac{84729414253 \sqrt{5 x+3} \sqrt{1-2 x}}{6553600000}+\frac{932023556783 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6553600000 \sqrt{10}} \]

[Out]

(84729414253*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6553600000 + (7702674023*(1 - 2*x)^(3/
2)*Sqrt[3 + 5*x])/1966080000 + (700243093*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4915200
00 - (63658463*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/16384000 - (5787133*(1 - 2*x)^(7/2
)*(3 + 5*x)^(3/2))/3072000 - (526103*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/768000 - (
(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/30 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(7/
2)*(245011 + 170940*x))/672000 + (932023556783*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(6553600000*Sqrt[10])

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Rubi [A]  time = 0.273722, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac{5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac{(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac{63658463 \sqrt{5 x+3} (1-2 x)^{7/2}}{16384000}+\frac{700243093 \sqrt{5 x+3} (1-2 x)^{5/2}}{491520000}+\frac{7702674023 \sqrt{5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac{84729414253 \sqrt{5 x+3} \sqrt{1-2 x}}{6553600000}+\frac{932023556783 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6553600000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(84729414253*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6553600000 + (7702674023*(1 - 2*x)^(3/
2)*Sqrt[3 + 5*x])/1966080000 + (700243093*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4915200
00 - (63658463*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/16384000 - (5787133*(1 - 2*x)^(7/2
)*(3 + 5*x)^(3/2))/3072000 - (526103*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/768000 - (
(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/30 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(7/
2)*(245011 + 170940*x))/672000 + (932023556783*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])
/(6553600000*Sqrt[10])

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Rubi in Sympy [A]  time = 24.0907, size = 197, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{7}{2}}}{30} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (128205 x + \frac{735033}{4}\right )}{504000} + \frac{526103 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{1920000} + \frac{5787133 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{19200000} + \frac{63658463 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{256000000} - \frac{700243093 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3072000000} - \frac{7702674023 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{4915200000} - \frac{84729414253 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6553600000} + \frac{932023556783 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{65536000000} \]

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)**(5/2),x)

[Out]

-(-2*x + 1)**(7/2)*(3*x + 2)**2*(5*x + 3)**(7/2)/30 - (-2*x + 1)**(7/2)*(5*x + 3
)**(7/2)*(128205*x + 735033/4)/504000 + 526103*(-2*x + 1)**(5/2)*(5*x + 3)**(7/2
)/1920000 + 5787133*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/19200000 + 63658463*sqrt(
-2*x + 1)*(5*x + 3)**(7/2)/256000000 - 700243093*sqrt(-2*x + 1)*(5*x + 3)**(5/2)
/3072000000 - 7702674023*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/4915200000 - 8472941425
3*sqrt(-2*x + 1)*sqrt(5*x + 3)/6553600000 + 932023556783*sqrt(10)*asin(sqrt(22)*
sqrt(5*x + 3)/11)/65536000000

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Mathematica [A]  time = 0.191912, size = 90, normalized size = 0.42 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (41287680000000 x^8+102445056000000 x^7+59625676800000 x^6-46327577600000 x^5-60250198784000 x^4-5712426076800 x^3+16445681555360 x^2+6397508631020 x-1496712721437\right )-19572494692443 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1376256000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-1496712721437 + 6397508631020*x + 164456815553
60*x^2 - 5712426076800*x^3 - 60250198784000*x^4 - 46327577600000*x^5 + 596256768
00000*x^6 + 102445056000000*x^7 + 41287680000000*x^8) - 19572494692443*Sqrt[10]*
ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1376256000000

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Maple [A]  time = 0.018, size = 189, normalized size = 0.9 \[{\frac{1}{2752512000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 825753600000000\,{x}^{8}\sqrt{-10\,{x}^{2}-x+3}+2048901120000000\,{x}^{7}\sqrt{-10\,{x}^{2}-x+3}+1192513536000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-926551552000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1205003975680000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-114248521536000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-29934254428740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2752512000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(825753600000000*x^8*(-10*x^2-x+3)^(
1/2)+2048901120000000*x^7*(-10*x^2-x+3)^(1/2)+1192513536000000*x^6*(-10*x^2-x+3)
^(1/2)-926551552000000*x^5*(-10*x^2-x+3)^(1/2)-1205003975680000*x^4*(-10*x^2-x+3
)^(1/2)-114248521536000*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)-29934254428740*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5052, size = 196, normalized size = 0.91 \[ -\frac{3}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} - \frac{1047}{1600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{111537}{224000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{526103}{384000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{526103}{7680000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{63658463}{12288000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{63658463}{245760000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{7702674023}{327680000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{932023556783}{131072000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{7702674023}{6553600000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/10*(-10*x^2 - x + 3)^(7/2)*x^2 - 1047/1600*(-10*x^2 - x + 3)^(7/2)*x - 111537
/224000*(-10*x^2 - x + 3)^(7/2) + 526103/384000*(-10*x^2 - x + 3)^(5/2)*x + 5261
03/7680000*(-10*x^2 - x + 3)^(5/2) + 63658463/12288000*(-10*x^2 - x + 3)^(3/2)*x
+ 63658463/245760000*(-10*x^2 - x + 3)^(3/2) + 7702674023/327680000*sqrt(-10*x^
2 - x + 3)*x - 932023556783/131072000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 7702
674023/6553600000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.219609, size = 124, normalized size = 0.57 \[ \frac{1}{2752512000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19572494692443 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2752512000000*sqrt(10)*(2*sqrt(10)*(41287680000000*x^8 + 102445056000000*x^7 +
59625676800000*x^6 - 46327577600000*x^5 - 60250198784000*x^4 - 5712426076800*x^
3 + 16445681555360*x^2 + 6397508631020*x - 1496712721437)*sqrt(5*x + 3)*sqrt(-2*
x + 1) + 19572494692443*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293703, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done