3.2397 \(\int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{121}{256} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{5/2}}{7680}+\frac{14641 \sqrt{5 x+3} (1-2 x)^{3/2}}{30720}+\frac{161051 \sqrt{5 x+3} \sqrt{1-2 x}}{102400}+\frac{1771561 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

[Out]

(161051*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 + (14641*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/30720 + (1331*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/7680 - (121*(1 - 2*x)^(7/2)*Sqr
t[3 + 5*x])/256 - (11*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/48 - ((1 - 2*x)^(7/2)*(3
+ 5*x)^(5/2))/12 + (1771561*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*Sqrt[10])

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Rubi [A]  time = 0.157384, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{121}{256} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{5/2}}{7680}+\frac{14641 \sqrt{5 x+3} (1-2 x)^{3/2}}{30720}+\frac{161051 \sqrt{5 x+3} \sqrt{1-2 x}}{102400}+\frac{1771561 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(161051*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 + (14641*(1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/30720 + (1331*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/7680 - (121*(1 - 2*x)^(7/2)*Sqr
t[3 + 5*x])/256 - (11*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/48 - ((1 - 2*x)^(7/2)*(3
+ 5*x)^(5/2))/12 + (1771561*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*Sqrt[10])

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Rubi in Sympy [A]  time = 13.7183, size = 144, normalized size = 0.9 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{30} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{300} + \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{4000} - \frac{1331 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{48000} - \frac{14641 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{76800} - \frac{161051 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{102400} + \frac{1771561 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1024000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*(5*x + 3)**(7/2)/30 + 11*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/30
0 + 121*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/4000 - 1331*sqrt(-2*x + 1)*(5*x + 3)**(5
/2)/48000 - 14641*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/76800 - 161051*sqrt(-2*x + 1)*
sqrt(5*x + 3)/102400 + 1771561*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024000

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Mathematica [A]  time = 0.0934344, size = 75, normalized size = 0.47 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (5120000 x^5+1280000 x^4-4905600 x^3-748640 x^2+1895020 x+96003\right )-5314683 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3072000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(96003 + 1895020*x - 748640*x^2 - 4905600*x^3 +
1280000*x^4 + 5120000*x^5) - 5314683*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
3072000

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Maple [A]  time = 0.007, size = 136, normalized size = 0.9 \[{\frac{1}{30} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{300} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{121}{4000} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{48000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{14641}{76800} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{161051}{102400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1771561\,\sqrt{10}}{2048000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(1-2*x)^(5/2)*(3+5*x)^(7/2)+11/300*(1-2*x)^(3/2)*(3+5*x)^(7/2)+121/4000*(3+
5*x)^(7/2)*(1-2*x)^(1/2)-1331/48000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-14641/76800*(3+5
*x)^(3/2)*(1-2*x)^(1/2)-161051/102400*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1771561/204800
0*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/
11)

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Maxima [A]  time = 1.49591, size = 134, normalized size = 0.84 \[ \frac{1}{6} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{1}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{121}{192} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{121}{3840} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14641}{5120} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1771561}{2048000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{14641}{102400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(-10*x^2 - x + 3)^(5/2)*x + 1/120*(-10*x^2 - x + 3)^(5/2) + 121/192*(-10*x^2
- x + 3)^(3/2)*x + 121/3840*(-10*x^2 - x + 3)^(3/2) + 14641/5120*sqrt(-10*x^2 -
x + 3)*x - 1771561/2048000*sqrt(10)*arcsin(-20/11*x - 1/11) + 14641/102400*sqrt
(-10*x^2 - x + 3)

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Fricas [A]  time = 0.224842, size = 104, normalized size = 0.65 \[ \frac{1}{6144000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (5120000 \, x^{5} + 1280000 \, x^{4} - 4905600 \, x^{3} - 748640 \, x^{2} + 1895020 \, x + 96003\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 5314683 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6144000*sqrt(10)*(2*sqrt(10)*(5120000*x^5 + 1280000*x^4 - 4905600*x^3 - 748640
*x^2 + 1895020*x + 96003)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 5314683*arctan(1/20*sqr
t(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)*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.266218, size = 427, normalized size = 2.67 \[ \frac{1}{76800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{9600000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{59}{1920000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{4000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/76800000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) -
318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/9600000*sqrt(5)*(2*(
4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*s
qrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)
)) - 59/1920000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)
*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3
))) - 1/4000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x
+ 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/400*sqrt(5)*(2*(20*x
+ 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)))