3.2466 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)}-\frac{185}{126} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{125}{54} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{173 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{189 \sqrt{7}} \]

[Out]

(-185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/126 + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(21*(2
+ 3*x)) + (125*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/54 - (173*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(189*Sqrt[7])

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Rubi [A]  time = 0.238545, antiderivative size = 115, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)}-\frac{185}{126} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{125}{54} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{173 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{189 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/126 + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(21*(2
+ 3*x)) + (125*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/54 - (173*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(189*Sqrt[7])

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Rubi in Sympy [A]  time = 23.1721, size = 100, normalized size = 0.87 \[ - \frac{185 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{126} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (3 x + 2\right )} + \frac{125 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{108} - \frac{173 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1323} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-185*sqrt(-2*x + 1)*sqrt(5*x + 3)/126 + sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(21*(3*x
+ 2)) + 125*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/108 - 173*sqrt(7)*atan(sqr
t(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/1323

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Mathematica [A]  time = 0.208791, size = 107, normalized size = 0.93 \[ \frac{-\frac{84 \sqrt{1-2 x} \sqrt{5 x+3} (525 x+352)}{3 x+2}-692 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+6125 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{10584} \]

Antiderivative was successfully verified.

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

[Out]

((-84*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(352 + 525*x))/(2 + 3*x) - 692*Sqrt[7]*ArcTan[
(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 6125*Sqrt[10]*ArcTan[(1 + 20*x)
/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/10584

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Maple [A]  time = 0.019, size = 146, normalized size = 1.3 \[{\frac{1}{21168+31752\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2076\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+18375\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1384\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +12250\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -44100\,x\sqrt{-10\,{x}^{2}-x+3}-29568\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10584*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2076*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x+18375*10^(1/2)*arcsin(20/11*x+1/11)*x+1384*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+12250*10^(1/2)*arcsin(20/11*x+1/11)
-44100*x*(-10*x^2-x+3)^(1/2)-29568*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3
*x)

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Maxima [A]  time = 1.50966, size = 101, normalized size = 0.88 \[ \frac{125}{216} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{173}{2646} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{25}{18} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{\sqrt{-10 \, x^{2} - x + 3}}{63 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/216*sqrt(10)*arcsin(20/11*x + 1/11) + 173/2646*sqrt(7)*arcsin(37/11*x/abs(3*
x + 2) + 20/11/abs(3*x + 2)) - 25/18*sqrt(-10*x^2 - x + 3) - 1/63*sqrt(-10*x^2 -
x + 3)/(3*x + 2)

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Fricas [A]  time = 0.228123, size = 161, normalized size = 1.4 \[ -\frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (525 \, x + 352\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 875 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 346 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{10584 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10584*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(525*x + 352)*sqrt(5*x + 3)*sqrt(-2*
x + 1) - 875*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(s
qrt(5*x + 3)*sqrt(-2*x + 1))) - 346*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(-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)/(2+3*x)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.326344, size = 377, normalized size = 3.28 \[ \frac{173}{26460} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{125}{216} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{18} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{63 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

173/26460*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)))) + 125/216*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)))) - 5/18
*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 22/63*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)