3.2534 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]

[Out]

(243*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/220 + (7*(2 + 3*x)^3*Sqrt[3 + 5*x]
)/(11*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(27269 + 11316*x))/7040 -
(184641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*Sqrt[10])

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Rubi [A]  time = 0.188224, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(243*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/220 + (7*(2 + 3*x)^3*Sqrt[3 + 5*x]
)/(11*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(27269 + 11316*x))/7040 -
(184641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*Sqrt[10])

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Rubi in Sympy [A]  time = 19.4104, size = 105, normalized size = 0.93 \[ \frac{243 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{220} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{1909575 x}{2} + \frac{18406575}{8}\right )}{66000} - \frac{184641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6400} + \frac{7 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/220 + sqrt(-2*x + 1)*sqrt(5*x + 3)
*(1909575*x/2 + 18406575/8)/66000 - 184641*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/
11)/6400 + 7*(3*x + 2)**3*sqrt(5*x + 3)/(11*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0961245, size = 69, normalized size = 0.61 \[ \frac{2031051 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (19008 x^3+78408 x^2+196614 x-312365\right )}{70400 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-312365 + 196614*x + 78408*x^2 + 19008*x^3) + 2031051*Sqrt[1
0 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(70400*Sqrt[1 - 2*x])

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Maple [A]  time = 0.019, size = 123, normalized size = 1.1 \[ -{\frac{1}{-140800+281600\,x} \left ( -380160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4062102\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1568160\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-2031051\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3932280\,x\sqrt{-10\,{x}^{2}-x+3}+6247300\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140800*(-380160*x^3*(-10*x^2-x+3)^(1/2)+4062102*10^(1/2)*arcsin(20/11*x+1/11)
*x-1568160*x^2*(-10*x^2-x+3)^(1/2)-2031051*10^(1/2)*arcsin(20/11*x+1/11)-3932280
*x*(-10*x^2-x+3)^(1/2)+6247300*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/
(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50258, size = 111, normalized size = 0.98 \[ \frac{27}{20} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{184641}{12800} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{999}{160} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2187}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \, \sqrt{-10 \, x^{2} - x + 3}}{88 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/20*sqrt(-10*x^2 - x + 3)*x^2 - 184641/12800*sqrt(5)*sqrt(2)*arcsin(20/11*x +
1/11) + 999/160*sqrt(-10*x^2 - x + 3)*x + 2187/128*sqrt(-10*x^2 - x + 3) - 2401/
88*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.238541, size = 107, normalized size = 0.95 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (19008 \, x^{3} + 78408 \, x^{2} + 196614 \, x - 312365\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2031051 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{140800 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/140800*sqrt(10)*(2*sqrt(10)*(19008*x^3 + 78408*x^2 + 196614*x - 312365)*sqrt(5
*x + 3)*sqrt(-2*x + 1) - 2031051*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))))/(2*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**4/((-2*x + 1)**(3/2)*sqrt(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.23141, size = 113, normalized size = 1. \[ -\frac{184641}{6400} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (594 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 93 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 5179 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 50776531 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4400000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-184641/6400*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/4400000*(594*(4*(8
*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) + 5179*sqrt(5))*(5*x + 3) - 50776531*
sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)