3.2528 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)}+\frac{32 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)}-\frac{25}{9} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{441 \sqrt{7}} \]

[Out]

(32*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[
1 - 2*x]*(2 + 3*x)) - (25*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 - (169*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(441*Sqrt[7])

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Rubi [A]  time = 0.244374, antiderivative size = 122, 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 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)}+\frac{32 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)}-\frac{25}{9} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{441 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(32*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[
1 - 2*x]*(2 + 3*x)) - (25*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 - (169*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(441*Sqrt[7])

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Rubi in Sympy [A]  time = 22.8858, size = 107, normalized size = 0.88 \[ \frac{32 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{147 \left (3 x + 2\right )} - \frac{25 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{18} - \frac{169 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3087} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32*sqrt(-2*x + 1)*sqrt(5*x + 3)/(147*(3*x + 2)) - 25*sqrt(10)*asin(sqrt(22)*sqrt
(5*x + 3)/11)/18 - 169*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/30
87 + 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2))

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Mathematica [A]  time = 0.199502, size = 110, normalized size = 0.9 \[ \frac{-\frac{84 \sqrt{1-2 x} \sqrt{5 x+3} (1091 x+725)}{6 x^2+x-2}-338 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-8575 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{12348} \]

Antiderivative was successfully verified.

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

[Out]

((-84*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(725 + 1091*x))/(-2 + x + 6*x^2) - 338*Sqrt[7]
*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 8575*Sqrt[10]*ArcTan[(1
+ 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/12348

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Maple [B]  time = 0.02, size = 198, normalized size = 1.6 \[{\frac{1}{ \left ( 24696+37044\,x \right ) \left ( -1+2\,x \right ) } \left ( 2028\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-51450\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+338\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-8575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-676\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +17150\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -91644\,x\sqrt{-10\,{x}^{2}-x+3}-60900\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12348*(2028*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-514
50*10^(1/2)*arcsin(20/11*x+1/11)*x^2+338*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x-8575*10^(1/2)*arcsin(20/11*x+1/11)*x-676*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+17150*10^(1/2)*arcsin(20/11*x+1/11)-91
644*x*(-10*x^2-x+3)^(1/2)-60900*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)
/(2+3*x)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50735, size = 139, normalized size = 1.14 \[ -\frac{25}{36} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{169}{6174} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{5455 \, x}{441 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{9784}{1323 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{189 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/36*sqrt(10)*arcsin(20/11*x + 1/11) + 169/6174*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 5455/441*x/sqrt(-10*x^2 - x + 3) + 9784/1323/sqrt(
-10*x^2 - x + 3) + 1/189/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.233575, size = 173, normalized size = 1.42 \[ -\frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (1091 \, x + 725\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1225 \, \sqrt{7} \sqrt{5}{\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 169 \, \sqrt{2}{\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12348 \,{\left (6 \, x^{2} + x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12348*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(1091*x + 725)*sqrt(5*x + 3)*sqrt(-2
*x + 1) + 1225*sqrt(7)*sqrt(5)*(6*x^2 + x - 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 169*sqrt(2)*(6*x^2 + x - 2)*arctan(1/14*
sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(6*x^2 + x - 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.362595, size = 386, normalized size = 3.16 \[ \frac{169}{61740} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{25}{36} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{121 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{245 \,{\left (2 \, x - 1\right )}} - \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{147 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

169/61740*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)))) - 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)))) - 121/24
5*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 22/147*sqrt(10)*((sqrt(2)*sq
rt(-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)