3.2417 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=180 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (2023*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(360*(2 + 3*x)^4) + (67187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2160*(2 + 3*x)^
3) + (2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^2) + (245529161*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(169344*(2 + 3*x)) - (104040277*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi [A]  time = 0.369315, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

(7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (2023*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(360*(2 + 3*x)^4) + (67187*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2160*(2 + 3*x)^
3) + (2347559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12096*(2 + 3*x)^2) + (245529161*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(169344*(2 + 3*x)) - (104040277*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi in Sympy [A]  time = 35.2568, size = 165, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{5}} + \frac{245529161 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{169344 \left (3 x + 2\right )} + \frac{2347559 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12096 \left (3 x + 2\right )^{2}} + \frac{67187 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2160 \left (3 x + 2\right )^{3}} + \frac{2023 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{360 \left (3 x + 2\right )^{4}} - \frac{104040277 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(1/2),x)

[Out]

7*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(15*(3*x + 2)**5) + 245529161*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(169344*(3*x + 2)) + 2347559*sqrt(-2*x + 1)*sqrt(5*x + 3)/(12096*(3
*x + 2)**2) + 67187*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2160*(3*x + 2)**3) + 2023*sqrt
(-2*x + 1)*sqrt(5*x + 3)/(360*(3*x + 2)**4) - 104040277*sqrt(7)*atan(sqrt(7)*sqr
t(-2*x + 1)/(7*sqrt(5*x + 3)))/43904

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Mathematica [A]  time = 0.113237, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (11048812245 x^4+29956486710 x^3+30475811404 x^2+13788819736 x+2341358496\right )}{(3 x+2)^5}-1560604155 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{1317120} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2341358496 + 13788819736*x + 30475811404*x^2 +
29956486710*x^3 + 11048812245*x^4))/(2 + 3*x)^5 - 1560604155*Sqrt[7]*ArcTan[(-2
0 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/1317120

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Maple [B]  time = 0.023, size = 298, normalized size = 1.7 \[{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 379226809665\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+426661359656\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x)

[Out]

1/1317120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(379226809665*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+1264089365550*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1685452487400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^3+154683371430*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+419390813940*x^3*
(-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+426661359656*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)+327
79018944*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.50292, size = 248, normalized size = 1.38 \[ \frac{104040277}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{45 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{637 \, \sqrt{-10 \, x^{2} - x + 3}}{120 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{67187 \, \sqrt{-10 \, x^{2} - x + 3}}{2160 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2347559 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{245529161 \, \sqrt{-10 \, x^{2} - x + 3}}{169344 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="maxima")

[Out]

104040277/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 49/4
5*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) +
637/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 67187/2
160*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 2347559/12096*sqrt(-10*
x^2 - x + 3)/(9*x^2 + 12*x + 4) + 245529161/169344*sqrt(-10*x^2 - x + 3)/(3*x +
2)

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Fricas [A]  time = 0.230638, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1560604155 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1317120 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="fricas")

[Out]

1/1317120*sqrt(7)*(2*sqrt(7)*(11048812245*x^4 + 29956486710*x^3 + 30475811404*x^
2 + 13788819736*x + 2341358496)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1560604155*(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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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)**6/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.494343, size = 594, normalized size = 3.3 \[ \frac{104040277}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1331 \,{\left (706299 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 493892560 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 156884295680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 24022907776000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1441374466560000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9408 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="giac")

[Out]

104040277/878080*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)))) + 1331/9408*(706299*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)))^9 + 49389
2560*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 + 156884295680*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^5 + 24022907776000*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 + 1441
374466560000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(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)^5