3.2491 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}-\frac{602 \sqrt{1-2 x} (3 x+2)^2}{9075 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1020 x+12199)}{242000}+\frac{8127 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(165*(3 + 5*x)^(3/2)) - (602*Sqrt[1 - 2*x]*(2 + 3
*x)^2)/(9075*Sqrt[3 + 5*x]) - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12199 + 1020*x))/2
42000 + (8127*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[10])

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Rubi [A]  time = 0.195979, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}-\frac{602 \sqrt{1-2 x} (3 x+2)^2}{9075 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1020 x+12199)}{242000}+\frac{8127 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(165*(3 + 5*x)^(3/2)) - (602*Sqrt[1 - 2*x]*(2 + 3
*x)^2)/(9075*Sqrt[3 + 5*x]) - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12199 + 1020*x))/2
42000 + (8127*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[10])

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Rubi in Sympy [A]  time = 18.7961, size = 105, normalized size = 0.93 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{602 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{9075 \sqrt{5 x + 3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{26775 x}{2} + \frac{1280895}{8}\right )}{453750} + \frac{8127 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**3/(165*(5*x + 3)**(3/2)) - 602*sqrt(-2*x + 1)*(3*x
+ 2)**2/(9075*sqrt(5*x + 3)) - sqrt(-2*x + 1)*sqrt(5*x + 3)*(26775*x/2 + 1280895
/8)/453750 + 8127*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20000

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Mathematica [A]  time = 0.174261, size = 65, normalized size = 0.58 \[ -\frac{\sqrt{1-2 x} \left (2940300 x^3+11712195 x^2+10891910 x+2953931\right )}{726000 (5 x+3)^{3/2}}-\frac{8127 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(2953931 + 10891910*x + 11712195*x^2 + 2940300*x^3))/(726000*(3
+ 5*x)^(3/2)) - (8127*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2000*Sqrt[10])

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Maple [A]  time = 0.024, size = 130, normalized size = 1.2 \[{\frac{1}{14520000} \left ( 73752525\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-58806000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-234243900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -217838200\,x\sqrt{-10\,{x}^{2}-x+3}-59078620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14520000*(73752525*10^(1/2)*arcsin(20/11*x+1/11)*x^2-58806000*x^3*(-10*x^2-x+3
)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-234243900*x^2*(-10*x^2-x+3)^(1/
2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-217838200*x*(-10*x^2-x+3)^(1/2)-590786
20*(-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 = 1.53342, size = 123, normalized size = 1.09 \[ \frac{8127}{40000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{81}{500} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4509}{10000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{20625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{32 \, \sqrt{-10 \, x^{2} - x + 3}}{9075 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8127/40000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/500*sqrt(-10*x^2 - x + 3)
*x - 4509/10000*sqrt(-10*x^2 - x + 3) - 2/20625*sqrt(-10*x^2 - x + 3)/(25*x^2 +
30*x + 9) - 32/9075*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.232392, size = 120, normalized size = 1.06 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2940300 \, x^{3} + 11712195 \, x^{2} + 10891910 \, x + 2953931\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2950101 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{14520000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/14520000*sqrt(10)*(2*sqrt(10)*(2940300*x^3 + 11712195*x^2 + 10891910*x + 2953
931)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 2950101*(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. \[ \int \frac{\left (3 x + 2\right )^{4}}{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270291, size = 238, normalized size = 2.11 \[ -\frac{27}{50000} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 131 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{18150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{8127}{20000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{267 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1512500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{801 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1134375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/50000*(12*sqrt(5)*(5*x + 3) + 131*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
/18150000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 8127
/20000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 267/1512500*sqrt(10)*(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/1134375*(801*sqrt(10)*(sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))^3