3.2418 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=209 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (497*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(108*(2 + 3*x)^5) + (21199*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4)
+ (1729615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^3) + (302171615*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(338688*(2 + 3*x)^2) + (31603880465*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(4741632*(2 + 3*x)) - (13391796605*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(175616*Sqrt[7])

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Rubi [A]  time = 0.448089, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (497*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(108*(2 + 3*x)^5) + (21199*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4)
+ (1729615*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^3) + (302171615*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(338688*(2 + 3*x)^2) + (31603880465*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(4741632*(2 + 3*x)) - (13391796605*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(175616*Sqrt[7])

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Rubi in Sympy [A]  time = 42.4404, size = 192, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{18 \left (3 x + 2\right )^{6}} + \frac{31603880465 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4741632 \left (3 x + 2\right )} + \frac{302171615 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{338688 \left (3 x + 2\right )^{2}} + \frac{1729615 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12096 \left (3 x + 2\right )^{3}} + \frac{21199 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{864 \left (3 x + 2\right )^{4}} + \frac{497 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{108 \left (3 x + 2\right )^{5}} - \frac{13391796605 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**7/(3+5*x)**(1/2),x)

[Out]

7*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(18*(3*x + 2)**6) + 31603880465*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(4741632*(3*x + 2)) + 302171615*sqrt(-2*x + 1)*sqrt(5*x + 3)/(338
688*(3*x + 2)**2) + 1729615*sqrt(-2*x + 1)*sqrt(5*x + 3)/(12096*(3*x + 2)**3) +
21199*sqrt(-2*x + 1)*sqrt(5*x + 3)/(864*(3*x + 2)**4) + 497*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(108*(3*x + 2)**5) - 13391796605*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7
*sqrt(5*x + 3)))/1229312

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Mathematica [A]  time = 0.158378, size = 112, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (31603880465 (3 x+2)^5+4230402610 (3 x+2)^4+678009080 (3 x+2)^3+116340112 (3 x+2)^2+20590976 (3 x+2)+4302592\right )}{(3 x+2)^6}-361578508335 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{66382848} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4302592 + 20590976*(2 + 3*x) + 116340112*(2 +
3*x)^2 + 678009080*(2 + 3*x)^3 + 4230402610*(2 + 3*x)^4 + 31603880465*(2 + 3*x)^
5))/(2 + 3*x)^6 - 361578508335*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqr
t[3 + 5*x])])/66382848

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Maple [B]  time = 0.022, size = 346, normalized size = 1.7 \[{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 29287859175135\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+36876342922048\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(29287859175135*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+117151436700540*7^(1/2)*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+195252394500900*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+11946266815770*x^5*(-10*x^2-x+3)^(1/2)+173557
684000800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+4035392
0114760*x^4*(-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))*x^2+54544410839520*x^3*(-10*x^2-x+3)^(1/2)[phone]
440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+36876342922048*
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)[phone]*(-10*x^2-x+3
)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 1.50612, size = 311, normalized size = 1.49 \[ \frac{13391796605}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{54 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{469 \, \sqrt{-10 \, x^{2} - x + 3}}{108 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{21199 \, \sqrt{-10 \, x^{2} - x + 3}}{864 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1729615 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{302171615 \, \sqrt{-10 \, x^{2} - x + 3}}{338688 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{31603880465 \, \sqrt{-10 \, x^{2} - x + 3}}{4741632 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),x, algorithm="maxima")

[Out]

13391796605/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) +
49/54*sqrt(-10*x^2 - x + 3)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64) + 469/108*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 7
20*x^2 + 240*x + 32) + 21199/864*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x
^2 + 96*x + 16) + 1729615/12096*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x +
8) + 302171615/338688*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 31603880465/474
1632*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.228641, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (853304772555 \, x^{5} + 2882422865340 \, x^{4} + 3896029345680 \, x^{3} + 2634024494432 \, x^{2} + 890768460368 \, x + 120549503808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 40175389815 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),x, algorithm="fricas")

[Out]

1/7375872*sqrt(7)*(2*sqrt(7)*(853304772555*x^5 + 2882422865340*x^4 + 38960293456
80*x^3 + 2634024494432*x^2 + 890768460368*x + 120549503808)*sqrt(5*x + 3)*sqrt(-
2*x + 1) + 40175389815*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 57
6*x + 64)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*
x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**(5/2)/(2+3*x)**7/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.595288, size = 676, normalized size = 3.23 \[ \frac{2678359321}{4917248} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{6655 \,{\left (20305527 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 17887837240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 7599643632000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1749282956467200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 210267345272320000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 10389680589926400000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{263424 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),x, algorithm="giac")

[Out]

2678359321/4917248*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)))) + 6655/263424*(20305527*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)))^11
+ 17887837240*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 7599643632000*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^7 + 1749282956467200*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 + 210267345272320000*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 + 103896805899
26400000*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)^6