3.2316 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{36 (3 x+2)}+\frac{4}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{793 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x])/(36*(2 + 3*x)) + (4*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (793*Arc
Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(108*Sqrt[7])

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Rubi [A]  time = 0.238451, 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 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{41 \sqrt{5 x+3} \sqrt{1-2 x}}{36 (3 x+2)}+\frac{4}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{793 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (41*Sqrt[1 - 2*x]*Sqrt[3 + 5*
x])/(36*(2 + 3*x)) + (4*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (793*Arc
Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(108*Sqrt[7])

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Rubi in Sympy [A]  time = 23.0922, size = 107, normalized size = 0.89 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6 \left (3 x + 2\right )^{2}} + \frac{41 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36 \left (3 x + 2\right )} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{27} - \frac{793 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{756} \]

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)**3,x)

[Out]

-(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(6*(3*x + 2)**2) + 41*sqrt(-2*x + 1)*sqrt(5*x +
3)/(36*(3*x + 2)) + 4*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/27 - 793*sqrt(7)
*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/756

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Mathematica [A]  time = 0.145377, size = 107, normalized size = 0.89 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} (135 x+76)}{(3 x+2)^2}-793 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+112 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1512} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(76 + 135*x))/(2 + 3*x)^2 - 793*Sqrt[7]*ArcTan[
(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 112*Sqrt[10]*ArcTan[(1 + 20*x)/
(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1512

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Maple [B]  time = 0.017, size = 191, normalized size = 1.6 \[{\frac{1}{1512\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 7137\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1008\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+9516\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1344\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+3172\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +448\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +5670\,x\sqrt{-10\,{x}^{2}-x+3}+3192\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^3,x)

[Out]

1/1512*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(7137*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^2+1008*10^(1/2)*arcsin(20/11*x+1/11)*x^2+9516*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1344*10^(1/2)*arcsin(20/11*x+1/
11)*x+3172*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+448*10^(1/
2)*arcsin(20/11*x+1/11)+5670*x*(-10*x^2-x+3)^(1/2)+3192*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.52243, size = 136, normalized size = 1.13 \[ \frac{2}{27} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{793}{1512} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{5}{9} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{29 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*sqrt(10)*arcsin(20/11*x + 1/11) + 793/1512*sqrt(7)*arcsin(37/11*x/abs(3*x +
2) + 20/11/abs(3*x + 2)) + 5/9*sqrt(-10*x^2 - x + 3) + 1/2*(-10*x^2 - x + 3)^(3
/2)/(9*x^2 + 12*x + 4) - 29/36*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.229103, size = 165, normalized size = 1.38 \[ \frac{\sqrt{7}{\left (16 \, \sqrt{10} \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (135 \, x + 76\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 793 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1512 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1512*sqrt(7)*(16*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)*(135*x + 76)*sqrt(5*x + 3)*sq
rt(-2*x + 1) + 793*(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)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324861, size = 437, normalized size = 3.64 \[ \frac{793}{15120} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{2}{27} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{55 \,{\left (5 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 2296 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{18 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

793/15120*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)))) + 2/27*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)))) - 55/18*(
5*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 - 2296*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(2
2))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2