3.2386 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx\)

Optimal. Leaf size=150 \[ \frac{1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac{7093 (5 x+3)^{3/2} \sqrt{1-2 x}}{21600}-\frac{390869 \sqrt{5 x+3} \sqrt{1-2 x}}{259200}+\frac{1922677 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{777600 \sqrt{10}}-\frac{98}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-390869*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/259200 + (7093*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/21600 + (181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/1080 + ((1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/12 + (1922677*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(777600*Sqrt[10]) - (
98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi [A]  time = 0.367241, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac{7093 (5 x+3)^{3/2} \sqrt{1-2 x}}{21600}-\frac{390869 \sqrt{5 x+3} \sqrt{1-2 x}}{259200}+\frac{1922677 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{777600 \sqrt{10}}-\frac{98}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-390869*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/259200 + (7093*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/21600 + (181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/1080 + ((1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/12 + (1922677*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(777600*Sqrt[10]) - (
98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi in Sympy [A]  time = 37.5714, size = 138, normalized size = 0.92 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{12} - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{432} + \frac{871 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{8640} + \frac{77269 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{259200} + \frac{1922677 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{7776000} - \frac{98 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/12 - 181*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/432
+ 871*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/8640 + 77269*sqrt(-2*x + 1)*sqrt(5*x + 3)/
259200 + 1922677*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/7776000 - 98*sqrt(7)*a
tan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/243

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Mathematica [A]  time = 0.19285, size = 110, normalized size = 0.73 \[ \frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (432000 x^3-607200 x^2+230940 x+59599\right )-3136000 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+1922677 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{15552000} \]

Antiderivative was successfully verified.

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

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(59599 + 230940*x - 607200*x^2 + 432000*x^3) - 3
136000*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 1922677*S
qrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/15552000

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Maple [A]  time = 0.014, size = 132, normalized size = 0.9 \[{\frac{1}{15552000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 25920000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36432000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3136000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1922677\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+3575940\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15552000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(25920000*x^3*(-10*x^2-x+3)^(1/2)-3643200
0*x^2*(-10*x^2-x+3)^(1/2)+3136000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))+1922677*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2
)+3575940*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51457, size = 132, normalized size = 0.88 \[ -\frac{1}{6} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{271}{1080} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{7093}{4320} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1922677}{15552000} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49}{243} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{135521}{259200} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-10*x^2 - x + 3)^(3/2)*x + 271/1080*(-10*x^2 - x + 3)^(3/2) + 7093/4320*sq
rt(-10*x^2 - x + 3)*x + 1922677/15552000*sqrt(10)*arcsin(20/11*x + 1/11) + 49/24
3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 135521/259200*sqrt
(-10*x^2 - x + 3)

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Fricas [A]  time = 0.229065, size = 135, normalized size = 0.9 \[ \frac{1}{15552000} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (432000 \, x^{3} - 607200 \, x^{2} + 230940 \, x + 59599\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 313600 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 1922677 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15552000*sqrt(10)*(6*sqrt(10)*(432000*x^3 - 607200*x^2 + 230940*x + 59599)*sqr
t(5*x + 3)*sqrt(-2*x + 1) + 313600*sqrt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x +
20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 1922677*arctan(1/20*sqrt(10)*(20*x + 1)/(s
qrt(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)**(3/2)/(2+3*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313823, size = 269, normalized size = 1.79 \[ \frac{49}{2430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{1296000} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 577 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 23769 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 390869 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1922677}{15552000} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/2430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22
)))) + 1/1296000*(12*(8*(36*sqrt(5)*(5*x + 3) - 577*sqrt(5))*(5*x + 3) + 23769*s
qrt(5))*(5*x + 3) - 390869*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 1922677/1555
2000*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))