3.2318 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac{1991 (5 x+3)^{3/2} \sqrt{1-2 x}}{224 (3 x+2)^2}-\frac{21901 \sqrt{5 x+3} \sqrt{1-2 x}}{3136 (3 x+2)}-\frac{240911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) + (3*(1 - 2*x)^(5/2)*(3 +
5*x)^(3/2))/(28*(2 + 3*x)^4) + (181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(168*(2 + 3
*x)^3) + (1991*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(224*(2 + 3*x)^2) - (240911*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi [A]  time = 0.213009, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac{1991 (5 x+3)^{3/2} \sqrt{1-2 x}}{224 (3 x+2)^2}-\frac{21901 \sqrt{5 x+3} \sqrt{1-2 x}}{3136 (3 x+2)}-\frac{240911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) + (3*(1 - 2*x)^(5/2)*(3 +
5*x)^(3/2))/(28*(2 + 3*x)^4) + (181*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(168*(2 + 3
*x)^3) + (1991*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(224*(2 + 3*x)^2) - (240911*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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Rubi in Sympy [A]  time = 16.8534, size = 138, normalized size = 0.91 \[ - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1176 \left (3 x + 2\right )^{3}} + \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{28 \left (3 x + 2\right )^{4}} + \frac{1991 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4704 \left (3 x + 2\right )^{2}} + \frac{21901 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3136 \left (3 x + 2\right )} - \frac{240911 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-181*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(1176*(3*x + 2)**3) + 3*(-2*x + 1)**(5/2)*(
5*x + 3)**(3/2)/(28*(3*x + 2)**4) + 1991*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(4704*(
3*x + 2)**2) + 21901*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3136*(3*x + 2)) - 240911*sqrt
(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/21952

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Mathematica [A]  time = 0.0917673, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (1705089 x^3+3485960 x^2+2381420 x+541680\right )}{(3 x+2)^4}-722733 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{131712} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(541680 + 2381420*x + 3485960*x^2 + 1705089*x^3
))/(2 + 3*x)^4 - 722733*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5
*x])])/131712

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Maple [B]  time = 0.019, size = 250, normalized size = 1.7 \[{\frac{1}{131712\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 58541373\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+48803440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+7583520\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/131712*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(58541373*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+156110328*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^3+156110328*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2+23871246*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+48803440*x^2*(-10*x^2-x+3)^(1/2)[phone]
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-
x+3)^(1/2)+7583520*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 1.51879, size = 212, normalized size = 1.4 \[ \frac{240911}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{9955}{2352} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{168 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{5973 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1568 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{73667 \, \sqrt{-10 \, x^{2} - x + 3}}{9408 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

240911/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 9955/23
52*sqrt(-10*x^2 - x + 3) + 1/4*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x
^2 + 96*x + 16) + 169/168*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) +
5973/1568*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 73667/9408*sqrt(-10*x^2
- x + 3)/(3*x + 2)

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Fricas [A]  time = 0.226024, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1705089 \, x^{3} + 3485960 \, x^{2} + 2381420 \, x + 541680\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 722733 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{131712 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/131712*sqrt(7)*(2*sqrt(7)*(1705089*x^3 + 3485960*x^2 + 2381420*x + 541680)*sqr
t(5*x + 3)*sqrt(-2*x + 1) + 722733*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arct
an(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 +
216*x^2 + 96*x + 16)

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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)*(3+5*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.390501, size = 512, normalized size = 3.39 \[ \frac{240911}{439040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (543 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 696920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 156094400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 11919936000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4704 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

240911/439040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))) - 1331/4704*(543*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)))^7 - 696920*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 - 156094400*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
)))^3 - 11919936000*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)^4