3.2351 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{7 (3 x+2)^3}+\frac{59 \sqrt{5 x+3} (1-2 x)^{3/2}}{28 (3 x+2)^2}+\frac{1947 \sqrt{5 x+3} \sqrt{1-2 x}}{56 (3 x+2)}-\frac{21417 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

[Out]

((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (59*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(28*(2 + 3*x)^2) + (1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(56*(2 + 3*x)) - (214
17*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Rubi [A]  time = 0.172655, 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 \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{7 (3 x+2)^3}+\frac{59 \sqrt{5 x+3} (1-2 x)^{3/2}}{28 (3 x+2)^2}+\frac{1947 \sqrt{5 x+3} \sqrt{1-2 x}}{56 (3 x+2)}-\frac{21417 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (59*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(28*(2 + 3*x)^2) + (1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(56*(2 + 3*x)) - (214
17*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Rubi in Sympy [A]  time = 13.584, size = 109, normalized size = 0.89 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{7 \left (3 x + 2\right )^{3}} + \frac{59 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{28 \left (3 x + 2\right )^{2}} + \frac{1947 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{56 \left (3 x + 2\right )} - \frac{21417 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{392} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(7*(3*x + 2)**3) + 59*(-2*x + 1)**(3/2)*sqrt(5*x
+ 3)/(28*(3*x + 2)**2) + 1947*sqrt(-2*x + 1)*sqrt(5*x + 3)/(56*(3*x + 2)) - 214
17*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/392

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Mathematica [A]  time = 0.0879928, size = 77, normalized size = 0.63 \[ \frac{1}{784} \left (\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (16847 x^2+23214 x+8032\right )}{(3 x+2)^3}-21417 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8032 + 23214*x + 16847*x^2))/(2 + 3*x)^3 - 214
17*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/784

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Maple [B]  time = 0.022, size = 202, normalized size = 1.7 \[{\frac{1}{784\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 578259\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1156518\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+771012\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+235858\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+171336\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +324996\,x\sqrt{-10\,{x}^{2}-x+3}+112448\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/784*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(578259*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^3+1156518*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2+771012*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x+235858*x^2*(-10*x^2-x+3)^(1/2)+171336*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))+324996*x*(-10*x^2-x+3)^(1/2)+112448*(-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 = 1.49975, size = 144, normalized size = 1.18 \[ \frac{21417}{784} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{9 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{161 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{16847 \, \sqrt{-10 \, x^{2} - x + 3}}{504 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21417/784*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/9*sqrt(-
10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 161/36*sqrt(-10*x^2 - x + 3)/(9*x
^2 + 12*x + 4) + 16847/504*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.223228, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (16847 \, x^{2} + 23214 \, x + 8032\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 21417 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/784*sqrt(7)*(2*sqrt(7)*(16847*x^2 + 23214*x + 8032)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 21417*(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((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.33786, size = 429, normalized size = 3.52 \[ \frac{21417}{7840} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{121 \,{\left (383 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 132160 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 13876800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{28 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21417/7840*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)))) + 121/28*(383*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 + 132160*sqrt(10)*(
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^3 + 13876800*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(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)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^3