3.2572 \(\int \frac{(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=164 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt{1-2 x}}-\frac{3315}{352} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac{1626211523 \sqrt{1-2 x} \sqrt{5 x+3}}{1126400}+\frac{1626211523 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

[Out]

(-1626211523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1126400 - (3315*Sqrt[1 - 2*x]*(2 + 3*x
)^2*(3 + 5*x)^(3/2))/352 - (123*(2 + 3*x)^3*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x])
+ ((2 + 3*x)^4*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2)*(22868329 + 10798680*x))/281600 + (1626211523*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(102400*Sqrt[10])

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Rubi [A]  time = 0.279846, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt{1-2 x}}-\frac{3315}{352} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac{1626211523 \sqrt{1-2 x} \sqrt{5 x+3}}{1126400}+\frac{1626211523 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^4*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]

[Out]

(-1626211523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1126400 - (3315*Sqrt[1 - 2*x]*(2 + 3*x
)^2*(3 + 5*x)^(3/2))/352 - (123*(2 + 3*x)^3*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x])
+ ((2 + 3*x)^4*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2)*(22868329 + 10798680*x))/281600 + (1626211523*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(102400*Sqrt[10])

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Rubi in Sympy [A]  time = 35.0517, size = 158, normalized size = 0.96 \[ - \frac{3337 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{224} - \frac{7779 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{128} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{32107012875 x}{8} + \frac{309488440275}{32}\right )}{5040000} + \frac{1626211523 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1024000} - \frac{123 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{14 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

-3337*sqrt(-2*x + 1)*(3*x + 2)**3*sqrt(5*x + 3)/224 - 7779*sqrt(-2*x + 1)*(3*x +
2)**2*sqrt(5*x + 3)/128 - sqrt(-2*x + 1)*sqrt(5*x + 3)*(32107012875*x/8 + 30948
8440275/32)/5040000 + 1626211523*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/102400
0 - 123*(3*x + 2)**4*sqrt(5*x + 3)/(14*sqrt(-2*x + 1)) + (3*x + 2)**4*(5*x + 3)*
*(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.190477, size = 84, normalized size = 0.51 \[ \frac{4878634569 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (15552000 x^5+83548800 x^4+236669040 x^3+633940524 x^2-2034703904 x+739060191\right )}{3072000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^4*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]

[Out]

(-10*Sqrt[3 + 5*x]*(739060191 - 2034703904*x + 633940524*x^2 + 236669040*x^3 + 8
3548800*x^4 + 15552000*x^5) + 4878634569*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[
5/11]*Sqrt[1 - 2*x]])/(3072000*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.019, size = 171, normalized size = 1. \[{\frac{1}{6144000\, \left ( -1+2\,x \right ) ^{2}} \left ( -311040000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1670976000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-4733380800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-19514538276\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-12678810480\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-14781203820\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4*(3+5*x)^(3/2)/(1-2*x)^(5/2),x)

[Out]

1/6144000*(-311040000*x^5*(-10*x^2-x+3)^(1/2)-1670976000*x^4*(-10*x^2-x+3)^(1/2)
[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-4733380800*x^3*(-10*x^2-x+3)^(1/2
)-19514538276*10^(1/2)*arcsin(20/11*x+1/11)*x-12678810480*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)-147
81203820*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+
3)^(1/2)

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Maxima [A]  time = 1.54612, size = 325, normalized size = 1.98 \[ \frac{81}{64} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1666460963}{2048000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{251559}{12800} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{10161}{1280} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{2079}{32} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{29403}{5120} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{43659}{640} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{34897797}{102400} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1323 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} + \frac{26411 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{491519 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

81/64*(-10*x^2 - x + 3)^(3/2)*x + 1666460963/2048000*sqrt(5)*sqrt(2)*arcsin(20/1
1*x + 1/11) + 251559/12800*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) + 10161/128
0*(-10*x^2 - x + 3)^(3/2) - 2079/32*sqrt(10*x^2 - 21*x + 8)*x + 29403/5120*sqrt(
-10*x^2 - x + 3)*x + 43659/640*sqrt(10*x^2 - 21*x + 8) - 34897797/102400*sqrt(-1
0*x^2 - x + 3) - 2401/96*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 10
29/8*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 1323/32*(-10*x^2 - x + 3)^(3/2)
/(2*x - 1) + 26411/192*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 491519/192*sqrt
(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.23128, size = 134, normalized size = 0.82 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (15552000 \, x^{5} + 83548800 \, x^{4} + 236669040 \, x^{3} + 633940524 \, x^{2} - 2034703904 \, x + 739060191\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 4878634569 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6144000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/6144000*sqrt(10)*(2*sqrt(10)*(15552000*x^5 + 83548800*x^4 + 236669040*x^3 + 6
33940524*x^2 - 2034703904*x + 739060191)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 48786345
69*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x +
1))))/(4*x^2 - 4*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((2+3*x)**4*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241724, size = 149, normalized size = 0.91 \[ \frac{1626211523}{1024000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 427 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 42657 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9855815 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 3252423046 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 53664980259 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{38400000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^4/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

1626211523/1024000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/38400000*(4*
(9*(12*(8*(36*sqrt(5)*(5*x + 3) + 427*sqrt(5))*(5*x + 3) + 42657*sqrt(5))*(5*x +
3) + 9855815*sqrt(5))*(5*x + 3) - 3252423046*sqrt(5))*(5*x + 3) + 53664980259*s
qrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2