3.2592 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}-\frac{1589 \sqrt{5 x+3} (3 x+2)^2}{726 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2380020 x+5735477)}{193600}+\frac{392283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

[Out]

(-1589*(2 + 3*x)^2*Sqrt[3 + 5*x])/(726*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^3*Sqrt[3 +
5*x])/(33*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5735477 + 2380020*x))
/193600 + (392283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi [A]  time = 0.191153, 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 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}-\frac{1589 \sqrt{5 x+3} (3 x+2)^2}{726 \sqrt{1-2 x}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2380020 x+5735477)}{193600}+\frac{392283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-1589*(2 + 3*x)^2*Sqrt[3 + 5*x])/(726*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^3*Sqrt[3 +
5*x])/(33*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5735477 + 2380020*x))
/193600 + (392283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi in Sympy [A]  time = 18.7808, size = 105, normalized size = 0.93 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{1785015 x}{2} + \frac{17206431}{8}\right )}{72600} + \frac{392283 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16000} - \frac{1589 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{726 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*sqrt(5*x + 3)*(1785015*x/2 + 17206431/8)/72600 + 392283*sqrt(10)
*asin(sqrt(22)*sqrt(5*x + 3)/11)/16000 - 1589*(3*x + 2)**2*sqrt(5*x + 3)/(726*sq
rt(-2*x + 1)) + 7*(3*x + 2)**3*sqrt(5*x + 3)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.147487, size = 74, normalized size = 0.65 \[ \frac{142398729 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2352240 x^3+14544684 x^2-61036064 x+21305631\right )}{5808000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(21305631 - 61036064*x + 14544684*x^2 + 2352240*x^3) + 142398
729*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5808000*(1 - 2
*x)^(3/2))

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Maple [A]  time = 0.021, size = 137, normalized size = 1.2 \[{\frac{1}{11616000\, \left ( -1+2\,x \right ) ^{2}} \left ( 569594916\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-47044800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-569594916\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-290893680\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-426112620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11616000*(569594916*10^(1/2)*arcsin(20/11*x+1/11)*x^2-47044800*x^3*(-10*x^2-x+
3)^(1/2)-569594916*10^(1/2)*arcsin(20/11*x+1/11)*x-290893680*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)-42
6112620*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3
)^(1/2)

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Maxima [A]  time = 1.53977, size = 123, normalized size = 1.09 \[ \frac{392283}{32000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{81}{80} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{11637}{1600} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{2401 \, \sqrt{-10 \, x^{2} - x + 3}}{264 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{55909 \, \sqrt{-10 \, x^{2} - x + 3}}{1452 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

392283/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/80*sqrt(-10*x^2 - x + 3
)*x - 11637/1600*sqrt(-10*x^2 - x + 3) + 2401/264*sqrt(-10*x^2 - x + 3)/(4*x^2 -
4*x + 1) + 55909/1452*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.235275, size = 120, normalized size = 1.06 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2352240 \, x^{3} + 14544684 \, x^{2} - 61036064 \, x + 21305631\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 142398729 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{11616000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11616000*sqrt(10)*(2*sqrt(10)*(2352240*x^3 + 14544684*x^2 - 61036064*x + 2130
5631)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 142398729*(4*x^2 - 4*x + 1)*arctan(1/20*sqr
t(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.255409, size = 113, normalized size = 1. \[ \frac{392283}{16000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9801 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 263 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 94936582 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1566381795 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{72600000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

392283/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/72600000*(4*(9801*
(12*sqrt(5)*(5*x + 3) + 263*sqrt(5))*(5*x + 3) - 94936582*sqrt(5))*(5*x + 3) + 1
566381795*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2