3.2265 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) - (11*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(84*(2 + 3*x)^2) + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) - (13
31*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi [A]  time = 0.171935, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) - (11*Sqrt[1 - 2*x]*(3 + 5*x)
^(3/2))/(84*(2 + 3*x)^2) + (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) - (13
31*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi in Sympy [A]  time = 13.6648, size = 109, normalized size = 0.89 \[ - \frac{121 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{392 \left (3 x + 2\right )} - \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{84 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (3 x + 2\right )^{3}} - \frac{1331 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2744} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-121*sqrt(-2*x + 1)*sqrt(5*x + 3)/(392*(3*x + 2)) - 11*sqrt(-2*x + 1)*(5*x + 3)*
*(3/2)/(84*(3*x + 2)**2) + sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(3*(3*x + 2)**3) - 13
31*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2744

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Mathematica [A]  time = 0.0953837, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (4223 x^2+4478 x+1152\right )}{(3 x+2)^3}-3993 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{16464} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1152 + 4478*x + 4223*x^2))/(2 + 3*x)^3 - 3993*
Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/16464

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Maple [B]  time = 0.017, size = 202, normalized size = 1.7 \[{\frac{1}{16464\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 107811\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+215622\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+143748\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+59122\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+31944\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +62692\,x\sqrt{-10\,{x}^{2}-x+3}+16128\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16464*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(107811*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^3+215622*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^2+143748*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x+59122*x^2*(-10*x^2-x+3)^(1/2)+31944*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))+62692*x*(-10*x^2-x+3)^(1/2)+16128*(-10*x^2-x+3)^(1/2))/(-10*
x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.50034, size = 163, normalized size = 1.34 \[ \frac{1331}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{55}{294} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{407 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 55/294*sqr
t(-10*x^2 - x + 3) - 1/21*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) +
33/196*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 407/1176*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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Fricas [A]  time = 0.222715, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (4223 \, x^{2} + 4478 \, x + 1152\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3993 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{16464 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16464*sqrt(7)*(2*sqrt(7)*(4223*x^2 + 4478*x + 1152)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 3993*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.353877, size = 429, normalized size = 3.52 \[ \frac{1331}{54880} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 2240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 235200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{588 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/54880*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)))) - 1331/588*(3*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)))^5 + 2240*sqrt(10)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22)))^3 - 235200*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))))/(((sqr
t(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)^3