3.2319 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{28291441 \sqrt{5 x+3} \sqrt{1-2 x}}{1185408 (3 x+2)}+\frac{270463 \sqrt{5 x+3} \sqrt{1-2 x}}{84672 (3 x+2)^2}+\frac{7723 \sqrt{5 x+3} \sqrt{1-2 x}}{15120 (3 x+2)^3}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{11988317 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(360*(2 + 3*x)^4) + (7723*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3)
+ (270463*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) + (28291441*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (11988317*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.370248, 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 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{28291441 \sqrt{5 x+3} \sqrt{1-2 x}}{1185408 (3 x+2)}+\frac{270463 \sqrt{5 x+3} \sqrt{1-2 x}}{84672 (3 x+2)^2}+\frac{7723 \sqrt{5 x+3} \sqrt{1-2 x}}{15120 (3 x+2)^3}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{11988317 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(360*(2 + 3*x)^4) + (7723*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3)
+ (270463*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) + (28291441*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (11988317*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 36.7183, size = 163, normalized size = 0.91 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{5}} + \frac{28291441 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1185408 \left (3 x + 2\right )} + \frac{270463 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84672 \left (3 x + 2\right )^{2}} + \frac{7723 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15120 \left (3 x + 2\right )^{3}} + \frac{41 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{360 \left (3 x + 2\right )^{4}} - \frac{11988317 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

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)**6,x)

[Out]

-(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(15*(3*x + 2)**5) + 28291441*sqrt(-2*x + 1)*sqr
t(5*x + 3)/(1185408*(3*x + 2)) + 270463*sqrt(-2*x + 1)*sqrt(5*x + 3)/(84672*(3*x
+ 2)**2) + 7723*sqrt(-2*x + 1)*sqrt(5*x + 3)/(15120*(3*x + 2)**3) + 41*sqrt(-2*
x + 1)*sqrt(5*x + 3)/(360*(3*x + 2)**4) - 11988317*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.101566, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (1273114845 x^4+3451770150 x^3+3511594796 x^2+1588955864 x+269759904\right )}{(3 x+2)^5}-179824755 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{9219840} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(269759904 + 1588955864*x + 3511594796*x^2 + 34
51770150*x^3 + 1273114845*x^4))/(2 + 3*x)^5 - 179824755*Sqrt[7]*ArcTan[(-20 - 37
*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/9219840

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Maple [B]  time = 0.017, size = 298, normalized size = 1.7 \[{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 43697415465\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+49162327144\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(43697415465*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+145658051550*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+194210735400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^3+17823607830*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+48324782100*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+49162327144*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)^5

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Maxima [A]  time = 1.51668, size = 267, normalized size = 1.48 \[ \frac{11988317}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{495385}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{239 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{8395 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2352 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{297231 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{3665849 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11988317/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4953
85/32928*sqrt(-10*x^2 - x + 3) + 1/5*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32) + 239/280*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 2
16*x^3 + 216*x^2 + 96*x + 16) + 8395/2352*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x
^2 + 36*x + 8) + 297231/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 36658
49/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.221963, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1273114845 \, x^{4} + 3451770150 \, x^{3} + 3511594796 \, x^{2} + 1588955864 \, x + 269759904\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 179824755 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{9219840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*sqrt(7)*(2*sqrt(7)*(1273114845*x^4 + 3451770150*x^3 + 3511594796*x^2 +
1588955864*x + 269759904)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 179824755*(243*x^5 + 8
10*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)**(3/2)*(3+5*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.459314, size = 594, normalized size = 3.3 \[ \frac{11988317}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (27021 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 52500560 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 18029240320 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 2768103296000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 166086197760000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{65856 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11988317/6146560*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)))) - 1331/65856*(27021*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 - 52500
560*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 - 18029240320*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 - 2768103296000*sqrt(10)*((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)))^3 - 1660861
97760000*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)^5