3.2276 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)}+\frac{215}{84} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{25}{9} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2119 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{252 \sqrt{7}} \]

[Out]

(215*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/84 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(84*(2
+ 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + (25*Sqrt[5/2]*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 + (2119*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(252*Sqrt[7])

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Rubi [A]  time = 0.306367, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)}+\frac{215}{84} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{25}{9} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2119 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{252 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(215*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/84 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(84*(2
+ 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + (25*Sqrt[5/2]*ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 + (2119*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(252*Sqrt[7])

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Rubi in Sympy [A]  time = 30.0313, size = 128, normalized size = 0.89 \[ \frac{215 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84} - \frac{59 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{84 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{6 \left (3 x + 2\right )^{2}} + \frac{25 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{18} + \frac{2119 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1764} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

215*sqrt(-2*x + 1)*sqrt(5*x + 3)/84 - 59*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(84*(3*
x + 2)) - sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(6*(3*x + 2)**2) + 25*sqrt(10)*asin(sq
rt(22)*sqrt(5*x + 3)/11)/18 + 2119*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5
*x + 3)))/1764

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Mathematica [A]  time = 0.164312, size = 112, normalized size = 0.78 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (700 x^2+1039 x+380\right )}{(3 x+2)^2}+2119 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+2450 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{3528} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(380 + 1039*x + 700*x^2))/(2 + 3*x)^2 + 2119*Sq
rt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 2450*Sqrt[10]*ArcT
an[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/3528

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Maple [A]  time = 0.017, size = 208, normalized size = 1.4 \[ -{\frac{1}{3528\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 19071\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-22050\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+25428\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-29400\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-29400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8476\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -9800\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -43638\,x\sqrt{-10\,{x}^{2}-x+3}-15960\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3528*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(19071*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^2-22050*10^(1/2)*arcsin(20/11*x+1/11)*x^2+25428*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-29400*10^(1/2)*arcsin(20/11
*x+1/11)*x-29400*x^2*(-10*x^2-x+3)^(1/2)+8476*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))-9800*10^(1/2)*arcsin(20/11*x+1/11)-43638*x*(-10*x^2-x+3
)^(1/2)-15960*(-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 = 1.50636, size = 136, normalized size = 0.94 \[ \frac{25}{36} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2119}{3528} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{20}{21} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{42 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{9 \, \sqrt{-10 \, x^{2} - x + 3}}{28 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/36*sqrt(10)*arcsin(20/11*x + 1/11) - 2119/3528*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 20/21*sqrt(-10*x^2 - x + 3) + 1/42*(-10*x^2 - x +
3)^(3/2)/(9*x^2 + 12*x + 4) + 9/28*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.22886, size = 188, normalized size = 1.31 \[ \frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (700 \, x^{2} + 1039 \, x + 380\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 700 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 2119 \, \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7056 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7056*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(700*x^2 + 1039*x + 380)*sqrt(5*x + 3)
*sqrt(-2*x + 1) + 700*sqrt(7)*sqrt(5)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqr
t(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 2119*sqrt(2)*(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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.385649, size = 463, normalized size = 3.22 \[ -\frac{2119}{35280} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{25}{36} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{5}{27} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{11 \,{\left (247 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 87640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{378 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2119/35280*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)))) + 25/36*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)))) + 5/27
*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 11/378*(247*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) - s
qrt(22)))^3 + 87640*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