3.2492 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (9405 x+5831)}{18150 \sqrt{5 x+3}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(165*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(5831 + 94
05*x))/(18150*Sqrt[3 + 5*x]) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10
])

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Rubi [A]  time = 0.125204, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^2}{165 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (9405 x+5831)}{18150 \sqrt{5 x+3}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(165*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(5831 + 94
05*x))/(18150*Sqrt[3 + 5*x]) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10
])

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Rubi in Sympy [A]  time = 11.6424, size = 80, normalized size = 0.95 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (\frac{47025 x}{4} + \frac{29155}{4}\right )}{45375 \sqrt{5 x + 3}} + \frac{81 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**2/(165*(5*x + 3)**(3/2)) - 2*sqrt(-2*x + 1)*(47025*
x/4 + 29155/4)/(45375*sqrt(5*x + 3)) + 81*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/1
1)/500

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Mathematica [A]  time = 0.148541, size = 60, normalized size = 0.71 \[ -\frac{\sqrt{1-2 x} \left (49005 x^2+60010 x+18373\right )}{18150 (5 x+3)^{3/2}}-\frac{81 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(18373 + 60010*x + 49005*x^2))/(18150*(3 + 5*x)^(3/2)) - (81*Arc
Sin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(50*Sqrt[10])

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Maple [A]  time = 0.019, size = 113, normalized size = 1.4 \[{\frac{1}{363000} \left ( 735075\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+882090\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-980100\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+264627\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1200200\,x\sqrt{-10\,{x}^{2}-x+3}-367460\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/363000*(735075*10^(1/2)*arcsin(20/11*x+1/11)*x^2+882090*10^(1/2)*arcsin(20/11*
x+1/11)*x-980100*x^2*(-10*x^2-x+3)^(1/2)+264627*10^(1/2)*arcsin(20/11*x+1/11)-12
00200*x*(-10*x^2-x+3)^(1/2)-367460*(-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 = 1.50234, size = 103, normalized size = 1.23 \[ \frac{81}{1000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{27}{250} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{4125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{602 \, \sqrt{-10 \, x^{2} - x + 3}}{45375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/1000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 27/250*sqrt(-10*x^2 - x + 3) -
2/4125*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 602/45375*sqrt(-10*x^2 - x +
3)/(5*x + 3)

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Fricas [A]  time = 0.23495, size = 113, normalized size = 1.35 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (49005 \, x^{2} + 60010 \, x + 18373\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 29403 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{363000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/363000*sqrt(10)*(2*sqrt(10)*(49005*x^2 + 60010*x + 18373)*sqrt(5*x + 3)*sqrt(
-2*x + 1) - 29403*(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. \[ \int \frac{\left (3 x + 2\right )^{3}}{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263259, size = 220, normalized size = 2.62 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{3630000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{27}{1250} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{81}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{201 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{302500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{603 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{226875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3630000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 27/
1250*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 81/500*sqrt(10)*arcsin(1/11*sqrt(22
)*sqrt(5*x + 3)) - 201/302500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt
(5*x + 3) + 1/226875*(603*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