3.2264 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{252 (3 x+2)}-\frac{10}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{4091 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{756 \sqrt{7}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2))/(6*(2 + 3*x)^2) - (10*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (4091
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(756*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.240807, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{252 (3 x+2)}-\frac{10}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{4091 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{756 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2))/(6*(2 + 3*x)^2) - (10*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (4091
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(756*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7076, size = 109, normalized size = 0.91 \[ - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{252 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2}} - \frac{10 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{27} - \frac{4091 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{5292} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107*sqrt(-2*x + 1)*sqrt(5*x + 3)/(252*(3*x + 2)) - sqrt(-2*x + 1)*(5*x + 3)**(3
/2)/(6*(3*x + 2)**2) - 10*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/27 - 4091*sqr
t(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/5292

_______________________________________________________________________________________

Mathematica [A]  time = 0.187447, size = 107, normalized size = 0.89 \[ \frac{-\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} (531 x+340)}{(3 x+2)^2}-4091 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-1960 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{10584} \]

Antiderivative was successfully verified.

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

[Out]

((-42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(340 + 531*x))/(2 + 3*x)^2 - 4091*Sqrt[7]*ArcT
an[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 1960*Sqrt[10]*ArcTan[(1 + 20
*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/10584

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 191, normalized size = 1.6 \[{\frac{1}{10584\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 36819\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-17640\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+49092\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-23520\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+16364\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -7840\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -22302\,x\sqrt{-10\,{x}^{2}-x+3}-14280\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10584*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(36819*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^2-17640*10^(1/2)*arcsin(20/11*x+1/11)*x^2+49092*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-23520*10^(1/2)*arcsin(20/11
*x+1/11)*x+16364*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-7840
*10^(1/2)*arcsin(20/11*x+1/11)-22302*x*(-10*x^2-x+3)^(1/2)-14280*(-10*x^2-x+3)^(
1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

_______________________________________________________________________________________

Maxima [A]  time = 1.50862, size = 136, normalized size = 1.13 \[ -\frac{5}{27} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{4091}{10584} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5}{63} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{103 \, \sqrt{-10 \, x^{2} - x + 3}}{252 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/27*sqrt(10)*arcsin(20/11*x + 1/11) + 4091/10584*sqrt(7)*arcsin(37/11*x/abs(3*
x + 2) + 20/11/abs(3*x + 2)) - 5/63*sqrt(-10*x^2 - x + 3) - 1/14*(-10*x^2 - x +
3)^(3/2)/(9*x^2 + 12*x + 4) - 103/252*sqrt(-10*x^2 - x + 3)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.232248, size = 165, normalized size = 1.38 \[ -\frac{\sqrt{7}{\left (280 \, \sqrt{10} \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (531 \, x + 340\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 4091 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{10584 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10584*sqrt(7)*(280*sqrt(10)*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(10)*(
20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(531*x + 340)*sqrt(5*x + 3
)*sqrt(-2*x + 1) - 4091*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))))/(9*x^2 + 12*x + 4)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.33591, size = 437, normalized size = 3.64 \[ \frac{4091}{105840} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{27} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (107 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 48440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{126 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4091/105840*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))) - 5/27*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/12
6*(107*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 + 48440*sqrt(10)*((sqrt(2)*sqrt(-1
0*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)^2