3.2519 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=122 \[ \frac{4 (5 x+3)^{5/2}}{77 \sqrt{1-2 x} (3 x+2)^2}-\frac{25 \sqrt{1-2 x} (5 x+3)^{3/2}}{1078 (3 x+2)^2}-\frac{75 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}-\frac{825 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

(-75*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) - (25*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(1078*(2 + 3*x)^2) + (4*(3 + 5*x)^(5/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^2)
- (825*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.174841, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{5/2}}{77 \sqrt{1-2 x} (3 x+2)^2}-\frac{25 \sqrt{1-2 x} (5 x+3)^{3/2}}{1078 (3 x+2)^2}-\frac{75 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}-\frac{825 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-75*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) - (25*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(1078*(2 + 3*x)^2) + (4*(3 + 5*x)^(5/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^2)
- (825*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 13.5226, size = 109, normalized size = 0.89 \[ - \frac{75 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1372 \left (3 x + 2\right )} - \frac{825 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{25 \left (5 x + 3\right )^{\frac{3}{2}}}{98 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{3 \left (5 x + 3\right )^{\frac{5}{2}}}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1372*(3*x + 2)) - 825*sqrt(7)*atan(sqrt(7)*sqr
t(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 - 25*(5*x + 3)**(3/2)/(98*sqrt(-2*x + 1)*(3*
x + 2)) + 3*(5*x + 3)**(5/2)/(14*sqrt(-2*x + 1)*(3*x + 2)**2)

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Mathematica [A]  time = 0.121606, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{5 x+3} \left (2550 x^2+2245 x+396\right )}{\sqrt{1-2 x} (3 x+2)^2}-825 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{19208} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(396 + 2245*x + 2550*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^2) - 825*
Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/19208

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Maple [B]  time = 0.022, size = 209, normalized size = 1.7 \[{\frac{1}{19208\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) } \left ( 14850\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+12375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-3300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-35700\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-3300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -31430\,x\sqrt{-10\,{x}^{2}-x+3}-5544\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19208*(14850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+12
375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-3300*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-35700*x^2*(-10*x^2-x+3)^(1/
2)-3300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-31430*x*(-10*
x^2-x+3)^(1/2)-5544*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(
-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50782, size = 193, normalized size = 1.58 \[ \frac{825}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2125 \, x}{2058 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{126 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{235}{1764 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

825/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2125/2058*
x/sqrt(-10*x^2 - x + 3) + 625/4116/sqrt(-10*x^2 - x + 3) - 1/126/(9*sqrt(-10*x^2
- x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 235/1764
/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.233351, size = 127, normalized size = 1.04 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (2550 \, x^{2} + 2245 \, x + 396\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 825 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{19208 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19208*sqrt(7)*(2*sqrt(7)*(2550*x^2 + 2245*x + 396)*sqrt(5*x + 3)*sqrt(-2*x +
1) - 825*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))/(18*x^3 + 15*x^2 - 4*x - 4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.37412, size = 382, normalized size = 3.13 \[ \frac{165}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{44 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1715 \,{\left (2 \, x - 1\right )}} - \frac{11 \,{\left (13 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 6280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

165/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22)))) - 44/1715*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 11/98*(13*sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 6280*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))
/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)
*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2