3.2421 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{9}{200} \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{5 x+3}}+\frac{651 \sqrt{5 x+3} (1-2 x)^{5/2}}{22000}+\frac{651 \sqrt{5 x+3} (1-2 x)^{3/2}}{8000}+\frac{21483 \sqrt{5 x+3} \sqrt{1-2 x}}{80000}+\frac{236313 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(7/2))/(275*Sqrt[3 + 5*x]) + (21483*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8
0000 + (651*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/8000 + (651*(1 - 2*x)^(5/2)*Sqrt[3 +
5*x])/22000 - (9*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/200 + (236313*ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]])/(80000*Sqrt[10])

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Rubi [A]  time = 0.166253, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{9}{200} \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{2 (1-2 x)^{7/2}}{275 \sqrt{5 x+3}}+\frac{651 \sqrt{5 x+3} (1-2 x)^{5/2}}{22000}+\frac{651 \sqrt{5 x+3} (1-2 x)^{3/2}}{8000}+\frac{21483 \sqrt{5 x+3} \sqrt{1-2 x}}{80000}+\frac{236313 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(7/2))/(275*Sqrt[3 + 5*x]) + (21483*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8
0000 + (651*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/8000 + (651*(1 - 2*x)^(5/2)*Sqrt[3 +
5*x])/22000 - (9*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/200 + (236313*ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]])/(80000*Sqrt[10])

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Rubi in Sympy [A]  time = 14.6321, size = 126, normalized size = 0.91 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{200} - \frac{2 \left (- 2 x + 1\right )^{\frac{7}{2}}}{275 \sqrt{5 x + 3}} + \frac{651 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{22000} + \frac{651 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{8000} + \frac{21483 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{80000} + \frac{236313 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{800000} \]

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)**(3/2),x)

[Out]

-9*(-2*x + 1)**(7/2)*sqrt(5*x + 3)/200 - 2*(-2*x + 1)**(7/2)/(275*sqrt(5*x + 3))
+ 651*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/22000 + 651*(-2*x + 1)**(3/2)*sqrt(5*x +
3)/8000 + 21483*sqrt(-2*x + 1)*sqrt(5*x + 3)/80000 + 236313*sqrt(10)*asin(sqrt(2
2)*sqrt(5*x + 3)/11)/800000

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Mathematica [A]  time = 0.173516, size = 70, normalized size = 0.51 \[ \frac{\frac{10 \sqrt{1-2 x} \left (144000 x^4-77600 x^3-112620 x^2+134625 x+79699\right )}{\sqrt{5 x+3}}-236313 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{800000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(79699 + 134625*x - 112620*x^2 - 77600*x^3 + 144000*x^4))/Sqr
t[3 + 5*x] - 236313*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/800000

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Maple [A]  time = 0.02, size = 133, normalized size = 1. \[{\frac{1}{1600000} \left ( 2880000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1552000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1181565\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-2252400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+708939\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2692500\,x\sqrt{-10\,{x}^{2}-x+3}+1593980\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1600000*(2880000*x^4*(-10*x^2-x+3)^(1/2)-1552000*x^3*(-10*x^2-x+3)^(1/2)+11815
65*10^(1/2)*arcsin(20/11*x+1/11)*x-2252400*x^2*(-10*x^2-x+3)^(1/2)+708939*10^(1/
2)*arcsin(20/11*x+1/11)+2692500*x*(-10*x^2-x+3)^(1/2)+1593980*(-10*x^2-x+3)^(1/2
))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49356, size = 147, normalized size = 1.07 \[ -\frac{18 \, x^{5}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{187 \, x^{4}}{50 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3691 \, x^{3}}{2000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{38187 \, x^{2}}{8000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{236313}{1600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{24773 \, x}{80000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{79699}{80000 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18/5*x^5/sqrt(-10*x^2 - x + 3) + 187/50*x^4/sqrt(-10*x^2 - x + 3) + 3691/2000*x
^3/sqrt(-10*x^2 - x + 3) - 38187/8000*x^2/sqrt(-10*x^2 - x + 3) - 236313/1600000
*sqrt(10)*arcsin(-20/11*x - 1/11) - 24773/80000*x/sqrt(-10*x^2 - x + 3) + 79699/
80000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.22333, size = 113, normalized size = 0.82 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (144000 \, x^{4} - 77600 \, x^{3} - 112620 \, x^{2} + 134625 \, x + 79699\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 236313 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1600000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1600000*sqrt(10)*(2*sqrt(10)*(144000*x^4 - 77600*x^3 - 112620*x^2 + 134625*x +
79699)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 236313*(5*x + 3)*arctan(1/20*sqrt(10)*(20
*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(5*x + 3)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.319635, size = 185, normalized size = 1.34 \[ \frac{1}{2000000} \,{\left (4 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 529 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16905 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 61545 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{236313}{800000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{31250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{15625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2000000*(4*(8*(36*sqrt(5)*(5*x + 3) - 529*sqrt(5))*(5*x + 3) + 16905*sqrt(5))*
(5*x + 3) + 61545*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 236313/800000*sqrt(10
)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/31250*sqrt(10)*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) + 242/15625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-
10*x + 5) - sqrt(22))