3.2490 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}-\frac{734 \sqrt{1-2 x} (3 x+2)^3}{9075 \sqrt{5 x+3}}+\frac{511 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (366420 x+938509)}{4840000}+\frac{462357 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) - (734*Sqrt[1 - 2*x]*(2 + 3
*x)^3)/(9075*Sqrt[3 + 5*x]) + (511*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/3025
0 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x))/4840000 + (462357*ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000*Sqrt[10])

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Rubi [A]  time = 0.264547, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}-\frac{734 \sqrt{1-2 x} (3 x+2)^3}{9075 \sqrt{5 x+3}}+\frac{511 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (366420 x+938509)}{4840000}+\frac{462357 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) - (734*Sqrt[1 - 2*x]*(2 + 3
*x)^3)/(9075*Sqrt[3 + 5*x]) + (511*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/3025
0 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x))/4840000 + (462357*ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000*Sqrt[10])

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Rubi in Sympy [A]  time = 25.8521, size = 133, normalized size = 0.94 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{734 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{9075 \sqrt{5 x + 3}} + \frac{511 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{30250} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{28855575 x}{4} + \frac{295630335}{16}\right )}{13612500} + \frac{462357 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{400000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**4/(165*(5*x + 3)**(3/2)) - 734*sqrt(-2*x + 1)*(3*x
+ 2)**3/(9075*sqrt(5*x + 3)) + 511*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/302
50 - sqrt(-2*x + 1)*sqrt(5*x + 3)*(28855575*x/4 + 295630335/16)/13612500 + 46235
7*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/400000

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Mathematica [A]  time = 0.197561, size = 70, normalized size = 0.49 \[ -\frac{\sqrt{1-2 x} \left (117612000 x^4+502791300 x^3+1030526145 x^2+795297410 x+199549721\right )}{14520000 (5 x+3)^{3/2}}-\frac{462357 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{40000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(199549721 + 795297410*x + 1030526145*x^2 + 502791300*x^3 + 1176
12000*x^4))/(14520000*(3 + 5*x)^(3/2)) - (462357*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]
])/(40000*Sqrt[10])

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Maple [A]  time = 0.021, size = 147, normalized size = 1. \[{\frac{1}{290400000} \left ( -2352240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-10055826000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-20610522900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -15905948200\,x\sqrt{-10\,{x}^{2}-x+3}-3990994420\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/290400000*(-2352240000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/1
1*x+1/11)*x^2-10055826000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/
11*x+1/11)*x-20610522900*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/1
1*x+1/11)-15905948200*x*(-10*x^2-x+3)^(1/2)-3990994420*(-10*x^2-x+3)^(1/2))*(1-2
*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.50438, size = 146, normalized size = 1.03 \[ -\frac{81}{250} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{462357}{800000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9963}{10000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{305343}{200000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{103125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{998 \, \sqrt{-10 \, x^{2} - x + 3}}{1134375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/250*sqrt(-10*x^2 - x + 3)*x^2 + 462357/800000*sqrt(5)*sqrt(2)*arcsin(20/11*x
+ 1/11) - 9963/10000*sqrt(-10*x^2 - x + 3)*x - 305343/200000*sqrt(-10*x^2 - x +
3) - 2/103125*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 998/1134375*sqrt(-10*
x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.237297, size = 127, normalized size = 0.89 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (117612000 \, x^{4} + 502791300 \, x^{3} + 1030526145 \, x^{2} + 795297410 \, x + 199549721\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 167835591 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{290400000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/290400000*sqrt(10)*(2*sqrt(10)*(117612000*x^4 + 502791300*x^3 + 1030526145*x^
2 + 795297410*x + 199549721)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 167835591*(25*x^2 +
30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(25*x
^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.281475, size = 255, normalized size = 1.8 \[ -\frac{27}{1000000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 75 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{90750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{462357}{400000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{333 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{7562500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{999 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{5671875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/1000000*(12*(8*sqrt(5)*(5*x + 3) + 75*sqrt(5))*(5*x + 3) + 7745*sqrt(5))*sqr
t(5*x + 3)*sqrt(-10*x + 5) - 1/90750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))^3/(5*x + 3)^(3/2) + 462357/400000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3)) - 333/7562500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) +
1/5671875*(999*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sq
rt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3