3.2459 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{57595 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{85 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}-\frac{43 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^4}-\frac{78045 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) - (43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(504*(2 + 3*x)^3) + (85*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (57
595*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (78045*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.295899, 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 \[ \frac{57595 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{85 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}-\frac{43 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^4}-\frac{78045 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) - (43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(504*(2 + 3*x)^3) + (85*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (57
595*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (78045*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 29.1534, size = 136, normalized size = 0.9 \[ \frac{57595 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{197568 \left (3 x + 2\right )} + \frac{85 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14112 \left (3 x + 2\right )^{2}} - \frac{43 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{504 \left (3 x + 2\right )^{3}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84 \left (3 x + 2\right )^{4}} - \frac{78045 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

57595*sqrt(-2*x + 1)*sqrt(5*x + 3)/(197568*(3*x + 2)) + 85*sqrt(-2*x + 1)*sqrt(5
*x + 3)/(14112*(3*x + 2)**2) - 43*sqrt(-2*x + 1)*sqrt(5*x + 3)/(504*(3*x + 2)**3
) + sqrt(-2*x + 1)*sqrt(5*x + 3)/(84*(3*x + 2)**4) - 78045*sqrt(7)*atan(sqrt(7)*
sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.132314, size = 82, normalized size = 0.54 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (172785 x^3+346760 x^2+226348 x+48240\right )}{(3 x+2)^4}-702405 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2765952} \]

Antiderivative was successfully verified.

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

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(48240 + 226348*x + 346760*x^2 + 172785*x^3))/
(2 + 3*x)^4 - 702405*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x]
)])/2765952

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Maple [B]  time = 0.023, size = 250, normalized size = 1.7 \[{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6321645\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2418990\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7492320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+4854640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1248720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3168872\,x\sqrt{-10\,{x}^{2}-x+3}+675360\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(6321645*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+16857720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x^3+16857720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^2+2418990*x^3*(-10*x^2-x+3)^(1/2)+7492320*7^(1/2)*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+4854640*x^2*(-10*x^2-x+3)^(1/2)+1248720*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3168872*x*(-10*x^2-x+3)^(1/
2)+675360*(-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 = 1.50738, size = 193, normalized size = 1.28 \[ \frac{78045}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{43 \, \sqrt{-10 \, x^{2} - x + 3}}{504 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{85 \, \sqrt{-10 \, x^{2} - x + 3}}{14112 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{57595 \, \sqrt{-10 \, x^{2} - x + 3}}{197568 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

78045/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/84*sq
rt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 43/504*sqrt(-10*x
^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 85/14112*sqrt(-10*x^2 - x + 3)/(9*x^2
+ 12*x + 4) + 57595/197568*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.225949, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (172785 \, x^{3} + 346760 \, x^{2} + 226348 \, x + 48240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 78045 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*sqrt(7)*(2*sqrt(7)*(172785*x^3 + 346760*x^2 + 226348*x + 48240)*sqrt(5*
x + 3)*sqrt(-2*x + 1) + 78045*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(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(-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)/(2+3*x)**5/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.414952, size = 512, normalized size = 3.39 \[ \frac{15609}{614656} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{605 \,{\left (129 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 132440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 21026880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 2510681600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{10976 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15609/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 605/10976*(129*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 + 132440*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 - 21026880*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 - 2510681600*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