3.2460 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{694229 \sqrt{1-2 x} \sqrt{5 x+3}}{921984 (3 x+2)}+\frac{6107 \sqrt{1-2 x} \sqrt{5 x+3}}{65856 (3 x+2)^2}-\frac{73 \sqrt{1-2 x} \sqrt{5 x+3}}{11760 (3 x+2)^3}-\frac{367 \sqrt{1-2 x} \sqrt{5 x+3}}{5880 (3 x+2)^4}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^5}-\frac{2664057 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^5) - (367*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x])/(5880*(2 + 3*x)^4) - (73*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11760*(2 + 3*x)^3) +
(6107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(65856*(2 + 3*x)^2) + (694229*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(921984*(2 + 3*x)) - (2664057*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(307328*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.375669, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{694229 \sqrt{1-2 x} \sqrt{5 x+3}}{921984 (3 x+2)}+\frac{6107 \sqrt{1-2 x} \sqrt{5 x+3}}{65856 (3 x+2)^2}-\frac{73 \sqrt{1-2 x} \sqrt{5 x+3}}{11760 (3 x+2)^3}-\frac{367 \sqrt{1-2 x} \sqrt{5 x+3}}{5880 (3 x+2)^4}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^5}-\frac{2664057 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^5) - (367*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x])/(5880*(2 + 3*x)^4) - (73*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11760*(2 + 3*x)^3) +
(6107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(65856*(2 + 3*x)^2) + (694229*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(921984*(2 + 3*x)) - (2664057*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(307328*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.858, size = 163, normalized size = 0.91 \[ \frac{694229 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{921984 \left (3 x + 2\right )} + \frac{6107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{65856 \left (3 x + 2\right )^{2}} - \frac{73 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11760 \left (3 x + 2\right )^{3}} - \frac{367 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5880 \left (3 x + 2\right )^{4}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{105 \left (3 x + 2\right )^{5}} - \frac{2664057 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2151296} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

694229*sqrt(-2*x + 1)*sqrt(5*x + 3)/(921984*(3*x + 2)) + 6107*sqrt(-2*x + 1)*sqr
t(5*x + 3)/(65856*(3*x + 2)**2) - 73*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11760*(3*x +
2)**3) - 367*sqrt(-2*x + 1)*sqrt(5*x + 3)/(5880*(3*x + 2)**4) + sqrt(-2*x + 1)*s
qrt(5*x + 3)/(105*(3*x + 2)**5) - 2664057*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7
*sqrt(5*x + 3)))/2151296

_______________________________________________________________________________________

Mathematica [A]  time = 0.136464, size = 87, normalized size = 0.48 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (93720915 x^4+253769850 x^3+257531412 x^2+115804328 x+19437408\right )}{(3 x+2)^5}-39960855 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{64538880} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(19437408 + 115804328*x + 257531412*x^2 + 25376
9850*x^3 + 93720915*x^4))/(2 + 3*x)^5 - 39960855*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*
Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/64538880

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 298, normalized size = 1.7 \[{\frac{1}{21512960\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3236829255\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3605439768\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21512960*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(3236829255*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+10789430850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+14385907800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^3+1312092810*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3552777900*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+
3605439768*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

_______________________________________________________________________________________

Maxima [A]  time = 1.5042, size = 248, normalized size = 1.38 \[ \frac{2664057}{4302592} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{105 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{367 \, \sqrt{-10 \, x^{2} - x + 3}}{5880 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{73 \, \sqrt{-10 \, x^{2} - x + 3}}{11760 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{6107 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{694229 \, \sqrt{-10 \, x^{2} - x + 3}}{921984 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2664057/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/10
5*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) -
367/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 73/117
60*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 6107/65856*sqrt(-10*x^2
- x + 3)/(9*x^2 + 12*x + 4) + 694229/921984*sqrt(-10*x^2 - x + 3)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.227344, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (93720915 \, x^{4} + 253769850 \, x^{3} + 257531412 \, x^{2} + 115804328 \, x + 19437408\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 13320285 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{21512960 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21512960*sqrt(7)*(2*sqrt(7)*(93720915*x^4 + 253769850*x^3 + 257531412*x^2 + 11
5804328*x + 19437408)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 13320285*(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)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.523214, size = 594, normalized size = 3.3 \[ \frac{2664057}{43025920} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{121 \,{\left (22017 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 28768880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 9856573440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 2123818368000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 133530503680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{153664 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2664057/43025920*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)))) - 121/153664*(22017*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 + 28768
880*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 - 9856573440*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 - 2123818368000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 13353050
3680000*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 + 2
80)^5