3.2275 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=137 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{5}{6} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{95}{72} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{155}{216} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{59 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

[Out]

(-95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/72 + (5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6 - (Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) + (155*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/216 - (59*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(27*Sqrt[
7])

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Rubi [A]  time = 0.305133, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{5}{6} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{95}{72} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{155}{216} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{59 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/72 + (5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6 - (Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)) + (155*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]])/216 - (59*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(27*Sqrt[
7])

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Rubi in Sympy [A]  time = 30.9433, size = 121, normalized size = 0.88 \[ \frac{5 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6} - \frac{95 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{72} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (3 x + 2\right )} + \frac{155 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{432} - \frac{59 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{189} \]

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)**2,x)

[Out]

5*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/6 - 95*sqrt(-2*x + 1)*sqrt(5*x + 3)/72 - sqrt(
-2*x + 1)*(5*x + 3)**(5/2)/(3*(3*x + 2)) + 155*sqrt(10)*asin(sqrt(22)*sqrt(5*x +
3)/11)/432 - 59*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/189

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Mathematica [A]  time = 0.220677, size = 112, normalized size = 0.82 \[ \frac{\frac{84 \sqrt{1-2 x} \sqrt{5 x+3} \left (300 x^2+135 x-46\right )}{3 x+2}-944 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+1085 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{6048} \]

Antiderivative was successfully verified.

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

[Out]

((84*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-46 + 135*x + 300*x^2))/(2 + 3*x) - 944*Sqrt[7
]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 1085*Sqrt[10]*ArcTan[(
1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/6048

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Maple [A]  time = 0.019, size = 163, normalized size = 1.2 \[{\frac{1}{12096+18144\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2832\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3255\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+25200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1888\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2170\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +11340\,x\sqrt{-10\,{x}^{2}-x+3}-3864\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

1/6048*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2832*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x+3255*10^(1/2)*arcsin(20/11*x+1/11)*x+25200*x^2*(-10*x^2-x+
3)^(1/2)+1888*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2170*10
^(1/2)*arcsin(20/11*x+1/11)+11340*x*(-10*x^2-x+3)^(1/2)-3864*(-10*x^2-x+3)^(1/2)
)/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.48584, size = 122, normalized size = 0.89 \[ \frac{25}{18} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{155}{864} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{59}{378} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{65}{216} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{\sqrt{-10 \, x^{2} - x + 3}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*sqrt(-10*x^2 - x + 3)*x + 155/864*sqrt(10)*arcsin(20/11*x + 1/11) + 59/378
*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 65/216*sqrt(-10*x^2
- x + 3) - 1/27*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.229127, size = 167, normalized size = 1.22 \[ \frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (300 \, x^{2} + 135 \, x - 46\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 155 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 472 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6048 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6048*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(300*x^2 + 135*x - 46)*sqrt(5*x + 3)*s
qrt(-2*x + 1) + 155*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 472*sqrt(2)*(3*x + 2)*arctan(1/14*sqrt(7)
*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(3*x + 2)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.354057, size = 394, normalized size = 2.88 \[ \frac{59}{3780} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{216} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 49 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{155}{864} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{27 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

59/3780*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
)))) + 1/216*(12*sqrt(5)*(5*x + 3) - 49*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) +
155/864*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)))) - 22/27*sqrt(1
0)*((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)