3.2254 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{625115 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{6005 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{794365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(504*(2 + 3*x)^3) + (6005*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) +
(625115*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (794365*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.295652, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{625115 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{6005 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{794365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(504*(2 + 3*x)^3) + (6005*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) +
(625115*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (794365*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 28.4386, size = 136, normalized size = 0.9 \[ \frac{625115 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{197568 \left (3 x + 2\right )} + \frac{6005 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14112 \left (3 x + 2\right )^{2}} + \frac{37 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{504 \left (3 x + 2\right )^{3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12 \left (3 x + 2\right )^{4}} - \frac{794365 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625115*sqrt(-2*x + 1)*sqrt(5*x + 3)/(197568*(3*x + 2)) + 6005*sqrt(-2*x + 1)*sqr
t(5*x + 3)/(14112*(3*x + 2)**2) + 37*sqrt(-2*x + 1)*sqrt(5*x + 3)/(504*(3*x + 2)
**3) - sqrt(-2*x + 1)*sqrt(5*x + 3)/(12*(3*x + 2)**4) - 794365*sqrt(7)*atan(sqrt
(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.115399, size = 82, normalized size = 0.54 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (1875345 x^3+3834760 x^2+2617388 x+594416\right )}{(3 x+2)^4}-7149285 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2765952} \]

Antiderivative was successfully verified.

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

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(594416 + 2617388*x + 3834760*x^2 + 1875345*x^
3))/(2 + 3*x)^4 - 7149285*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 +
5*x])])/2765952

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Maple [B]  time = 0.019, size = 250, normalized size = 1.7 \[{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 64343565\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+53686640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+8321824\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(64343565*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+171582840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^3+171582840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2+26254830*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+53686640*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)+8321824*(-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 = 1.5358, size = 212, normalized size = 1.4 \[ \frac{794365}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{32825}{16464} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{185 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{19695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{242905 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

794365/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32825/
16464*sqrt(-10*x^2 - x + 3) + 3/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 2
16*x^2 + 96*x + 16) + 185/392*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x +
8) + 19695/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 242905/65856*sqrt(
-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.222035, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1875345 \, x^{3} + 3834760 \, x^{2} + 2617388 \, x + 594416\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 794365 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*sqrt(7)*(2*sqrt(7)*(1875345*x^3 + 3834760*x^2 + 2617388*x + 594416)*sqr
t(5*x + 3)*sqrt(-2*x + 1) + 794365*(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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.401375, size = 504, normalized size = 3.34 \[ \frac{121}{614656} \, \sqrt{5}{\left (1313 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{280 \, \sqrt{2}{\left (1313 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 1578920 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 374767680 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - \frac{28822976000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{115291904000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/614656*sqrt(5)*(1313*sqrt(70)*sqrt(2)*(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)))) - 280*sqrt(2)*(1313*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 1578920
*((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 - 374767680*((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 - 28822
976000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 115291904000*sqrt(5*
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 28
0)^4)