3.2395 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac{12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac{132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac{1456961 \sqrt{5 x+3} (1-2 x)^{7/2}}{819200}+\frac{16026571 \sqrt{5 x+3} (1-2 x)^{5/2}}{24576000}+\frac{176292281 \sqrt{5 x+3} (1-2 x)^{3/2}}{98304000}+\frac{1939215091 \sqrt{5 x+3} \sqrt{1-2 x}}{327680000}+\frac{21331366001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{327680000 \sqrt{10}} \]

[Out]

(1939215091*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/327680000 + (176292281*(1 - 2*x)^(3/2)*
Sqrt[3 + 5*x])/98304000 + (16026571*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24576000 - (1
456961*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/819200 - (132451*(1 - 2*x)^(7/2)*(3 + 5*x)
^(3/2))/153600 - (12041*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/38400 - (999*(1 - 2*x)^
(7/2)*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80
+ (21331366001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(327680000*Sqrt[10])

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Rubi [A]  time = 0.255442, antiderivative size = 209, 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 \[ -\frac{3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac{12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac{132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac{1456961 \sqrt{5 x+3} (1-2 x)^{7/2}}{819200}+\frac{16026571 \sqrt{5 x+3} (1-2 x)^{5/2}}{24576000}+\frac{176292281 \sqrt{5 x+3} (1-2 x)^{3/2}}{98304000}+\frac{1939215091 \sqrt{5 x+3} \sqrt{1-2 x}}{327680000}+\frac{21331366001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{327680000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1939215091*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/327680000 + (176292281*(1 - 2*x)^(3/2)*
Sqrt[3 + 5*x])/98304000 + (16026571*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24576000 - (1
456961*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/819200 - (132451*(1 - 2*x)^(7/2)*(3 + 5*x)
^(3/2))/153600 - (12041*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/38400 - (999*(1 - 2*x)^
(7/2)*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80
+ (21331366001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(327680000*Sqrt[10])

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Rubi in Sympy [A]  time = 20.8089, size = 190, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (9 x + 6\right )}{80} - \frac{999 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{11200} + \frac{12041 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{96000} - \frac{132451 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{384000} - \frac{1456961 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1228800} - \frac{16026571 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{4915200} + \frac{176292281 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{98304000} + \frac{1939215091 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{327680000} + \frac{21331366001 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3276800000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)*(5*x + 3)**(7/2)*(9*x + 6)/80 - 999*(-2*x + 1)**(7/2)*(5*x +
3)**(7/2)/11200 + 12041*(-2*x + 1)**(5/2)*(5*x + 3)**(7/2)/96000 - 132451*(-2*x
+ 1)**(5/2)*(5*x + 3)**(5/2)/384000 - 1456961*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)
/1228800 - 16026571*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/4915200 + 176292281*(-2*x +
1)**(3/2)*sqrt(5*x + 3)/98304000 + 1939215091*sqrt(-2*x + 1)*sqrt(5*x + 3)/32768
0000 + 21331366001*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/3276800000

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Mathematica [A]  time = 0.15558, size = 85, normalized size = 0.41 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (774144000000 x^7+1362124800000 x^6+97008640000 x^5-1013681408000 x^4-413675529600 x^3+252700365920 x^2+169330465940 x-22414998339\right )-447958686021 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{68812800000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-22414998339 + 169330465940*x + 252700365920*x^
2 - 413675529600*x^3 - 1013681408000*x^4 + 97008640000*x^5 + 1362124800000*x^6 +
774144000000*x^7) - 447958686021*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/688
12800000

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Maple [A]  time = 0.014, size = 172, normalized size = 0.8 \[{\frac{1}{137625600000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 15482880000000\,{x}^{7}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-20273628160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-8273510592000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-448299966780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/137625600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15482880000000*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)-
20273628160000*x^4*(-10*x^2-x+3)^(1/2)-8273510592000*x^3*(-10*x^2-x+3)^(1/2)+505
4007318400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)+33
86609318800*x*(-10*x^2-x+3)^(1/2)-448299966780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3
)^(1/2)

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Maxima [A]  time = 1.50217, size = 173, normalized size = 0.83 \[ -\frac{9}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{1839}{11200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{12041}{19200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{12041}{384000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{1456961}{614400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1456961}{12288000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{176292281}{16384000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21331366001}{6553600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{176292281}{327680000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/80*(-10*x^2 - x + 3)^(7/2)*x - 1839/11200*(-10*x^2 - x + 3)^(7/2) + 12041/192
00*(-10*x^2 - x + 3)^(5/2)*x + 12041/384000*(-10*x^2 - x + 3)^(5/2) + 1456961/61
4400*(-10*x^2 - x + 3)^(3/2)*x + 1456961/12288000*(-10*x^2 - x + 3)^(3/2) + 1762
92281/16384000*sqrt(-10*x^2 - x + 3)*x - 21331366001/6553600000*sqrt(10)*arcsin(
-20/11*x - 1/11) + 176292281/327680000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222762, size = 117, normalized size = 0.56 \[ \frac{1}{137625600000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (774144000000 \, x^{7} + 1362124800000 \, x^{6} + 97008640000 \, x^{5} - 1013681408000 \, x^{4} - 413675529600 \, x^{3} + 252700365920 \, x^{2} + 169330465940 \, x - 22414998339\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 447958686021 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/137625600000*sqrt(10)*(2*sqrt(10)*(774144000000*x^7 + 1362124800000*x^6 + 9700
8640000*x^5 - 1013681408000*x^4 - 413675529600*x^3 + 252700365920*x^2 + 16933046
5940*x - 22414998339)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 447958686021*arctan(1/20*sq
rt(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)**2*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.295485, 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)^2*(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

Done