3.2586 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=108 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{407 \sqrt{5 x+3}}{98 \sqrt{1-2 x}}+\frac{25}{6} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{147 \sqrt{7}} \]

[Out]

(-407*Sqrt[3 + 5*x])/(98*Sqrt[1 - 2*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/
2)) + (25*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/6 + (2*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(147*Sqrt[7])

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Rubi [A]  time = 0.242522, antiderivative size = 108, 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 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{407 \sqrt{5 x+3}}{98 \sqrt{1-2 x}}+\frac{25}{6} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{147 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-407*Sqrt[3 + 5*x])/(98*Sqrt[1 - 2*x]) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/
2)) + (25*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/6 + (2*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(147*Sqrt[7])

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Rubi in Sympy [A]  time = 23.0889, size = 99, normalized size = 0.92 \[ \frac{25 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{12} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1029} - \frac{407 \sqrt{5 x + 3}}{98 \sqrt{- 2 x + 1}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/12 + 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/1029 - 407*sqrt(5*x + 3)/(98*sqrt(-2*x + 1)) + 11*(5*x
+ 3)**(3/2)/(21*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.244243, size = 100, normalized size = 0.93 \[ \frac{\frac{308 \sqrt{5 x+3} (292 x-69)}{(1-2 x)^{3/2}}+8 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+8575 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{8232} \]

Antiderivative was successfully verified.

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

[Out]

((308*Sqrt[3 + 5*x]*(-69 + 292*x))/(1 - 2*x)^(3/2) + 8*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])])/8232

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Maple [B]  time = 0.02, size = 191, normalized size = 1.8 \[ -{\frac{1}{8232\, \left ( -1+2\,x \right ) ^{2}} \left ( 32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-34300\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+34300\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+8\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -8575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -89936\,x\sqrt{-10\,{x}^{2}-x+3}+21252\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8232*(32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-34300
*10^(1/2)*arcsin(20/11*x+1/11)*x^2-32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x+34300*10^(1/2)*arcsin(20/11*x+1/11)*x+8*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-8575*10^(1/2)*arcsin(20/11*x+1/11)-89936*x
*(-10*x^2-x+3)^(1/2)+21252*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+
2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52003, size = 220, normalized size = 2.04 \[ -\frac{12233125 \, x^{2}}{3557763 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625 \, x^{3}}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{25}{24} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{1029} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2446625}{7115526} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{12021894385 \, x}{697321548 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{16029625 \, x^{2}}{117612 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{6953014391}{697321548 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{12465295 \, x}{205821 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2681981}{274428 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12233125/3557763*x^2/sqrt(-10*x^2 - x + 3) + 625/6*x^3/(-10*x^2 - x + 3)^(3/2)
+ 25/24*sqrt(10)*arcsin(20/11*x + 1/11) - 1/1029*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) - 2446625/7115526*sqrt(-10*x^2 - x + 3) - 12021894385
/697321548*x/sqrt(-10*x^2 - x + 3) + 16029625/117612*x^2/(-10*x^2 - x + 3)^(3/2)
- 6953014391/697321548/sqrt(-10*x^2 - x + 3) + 12465295/205821*x/(-10*x^2 - x +
3)^(3/2) + 2681981/274428/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.232996, size = 181, normalized size = 1.68 \[ \frac{\sqrt{7} \sqrt{2}{\left (22 \, \sqrt{7} \sqrt{2}{\left (292 \, x - 69\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1225 \, \sqrt{7} \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 4 \, \sqrt{2}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{8232 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8232*sqrt(7)*sqrt(2)*(22*sqrt(7)*sqrt(2)*(292*x - 69)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 1225*sqrt(7)*sqrt(5)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x
+ 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 4*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/14*s
qrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*x + 1)

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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)**(5/2)/(2+3*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282072, size = 243, normalized size = 2.25 \[ -\frac{1}{10290} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{25}{24} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{11 \,{\left (292 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1221 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{7350 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10290*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(2
2)))) + 25/24*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)))) + 11/7350
*(292*sqrt(5)*(5*x + 3) - 1221*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^
2