3.2377 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/18 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(6*(2 + 3*
x)^2) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(4*(2 + 3*x)) + (118*Sqrt[2/5]*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (155*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/108

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Rubi [A]  time = 0.303739, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/18 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(6*(2 + 3*
x)^2) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(4*(2 + 3*x)) + (118*Sqrt[2/5]*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (155*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/108

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Rubi in Sympy [A]  time = 30.3586, size = 128, normalized size = 0.89 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{6 \left (3 x + 2\right )^{2}} + \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )} + \frac{19 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{18} + \frac{118 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{135} - \frac{155 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{108} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(6*(3*x + 2)**2) + 5*(-2*x + 1)**(3/2)*sqrt(5*x
+ 3)/(4*(3*x + 2)) + 19*sqrt(-2*x + 1)*sqrt(5*x + 3)/18 + 118*sqrt(10)*asin(sqr
t(22)*sqrt(5*x + 3)/11)/135 - 155*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*
x + 3)))/108

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Mathematica [A]  time = 0.172162, size = 112, normalized size = 0.78 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (48 x^2+435 x+236\right )}{(3 x+2)^2}-775 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+472 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1080} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(236 + 435*x + 48*x^2))/(2 + 3*x)^2 - 775*Sqrt[
7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 472*Sqrt[10]*ArcTan[(
1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1080

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Maple [A]  time = 0.017, size = 208, normalized size = 1.4 \[{\frac{1}{1080\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6975\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4248\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+9300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5664\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1888\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +13050\,x\sqrt{-10\,{x}^{2}-x+3}+7080\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1080*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6975*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^2+4248*10^(1/2)*arcsin(20/11*x+1/11)*x^2+9300*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+5664*10^(1/2)*arcsin(20/11*x+1/
11)*x+1440*x^2*(-10*x^2-x+3)^(1/2)+3100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))+1888*10^(1/2)*arcsin(20/11*x+1/11)+13050*x*(-10*x^2-x+3)^(1/2
)+7080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.51056, size = 136, normalized size = 0.94 \[ \frac{59}{135} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{155}{216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{13}{9} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

59/135*sqrt(10)*arcsin(20/11*x + 1/11) + 155/216*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 13/9*sqrt(-10*x^2 - x + 3) + 7/6*(-10*x^2 - x + 3)^
(3/2)/(9*x^2 + 12*x + 4) - 49/36*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.226926, size = 180, normalized size = 1.25 \[ \frac{\sqrt{5}{\left (155 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{5}{\left (48 \, x^{2} + 435 \, x + 236\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 472 \, \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1080 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1080*sqrt(5)*(155*sqrt(7)*sqrt(5)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x
+ 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(5)*(48*x^2 + 435*x + 236)*sqrt(5
*x + 3)*sqrt(-2*x + 1) + 472*sqrt(2)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqrt
(2)*(20*x + 1)/(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)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.400309, size = 463, normalized size = 3.22 \[ \frac{31}{432} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{59}{135} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{135} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{77 \,{\left (17 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 13720 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{54 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

31/432*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)
))) + 59/135*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)))) + 4/135*sq
rt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 77/54*(17*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)))^3 - 13720*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)^2