3.2278 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{\sqrt{1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}-\frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}-\frac{605 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{6655 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(-6655*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (605*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(4704*(2 + 3*x)^2) - (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*
x)^3) + (Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(4*(2 + 3*x)^4) - (73205*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.220904, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}-\frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}-\frac{605 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{6655 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-6655*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (605*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(4704*(2 + 3*x)^2) - (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*
x)^3) + (Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(4*(2 + 3*x)^4) - (73205*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 17.1048, size = 136, normalized size = 0.9 \[ - \frac{6655 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )} - \frac{605 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{4704 \left (3 x + 2\right )^{2}} - \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{168 \left (3 x + 2\right )^{3}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{4 \left (3 x + 2\right )^{4}} - \frac{73205 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

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)**5,x)

[Out]

-6655*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21952*(3*x + 2)) - 605*sqrt(-2*x + 1)*(5*x +
3)**(3/2)/(4704*(3*x + 2)**2) - 11*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(168*(3*x +
2)**3) + sqrt(-2*x + 1)*(5*x + 3)**(7/2)/(4*(3*x + 2)**4) - 73205*sqrt(7)*atan(s
qrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.125978, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{921984} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(105552 + 654436*x + 1285720*x^2 + 814395*x^3))
/(2 + 3*x)^4 - 219615*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x
])])/921984

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Maple [B]  time = 0.017, size = 250, normalized size = 1.7 \[{\frac{1}{921984\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 17788815\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+18000080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3513840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +9162104\,x\sqrt{-10\,{x}^{2}-x+3}+1477728\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^5,x)

[Out]

1/921984*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(17788815*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+47436840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^3+47436840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^2+11401530*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+18000080*x^2*(-10*x^2-x+3)^(1/2)+3513840*7^
(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+9162104*x*(-10*x^2-x+3)
^(1/2)+1477728*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 1.53154, size = 212, normalized size = 1.4 \[ \frac{73205}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3025}{16464} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{84 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{125 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1815 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{22385 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

73205/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3025/16
464*sqrt(-10*x^2 - x + 3) + 1/84*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216
*x^2 + 96*x + 16) - 125/1176*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8
) + 1815/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 22385/65856*sqrt(-10
*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.223365, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (814395 \, x^{3} + 1285720 \, x^{2} + 654436 \, x + 105552\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 219615 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{921984 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/921984*sqrt(7)*(2*sqrt(7)*(814395*x^3 + 1285720*x^2 + 654436*x + 105552)*sqrt(
5*x + 3)*sqrt(-2*x + 1) + 219615*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan
(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 + 2
16*x^2 + 96*x + 16)

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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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.413823, size = 512, normalized size = 3.39 \[ \frac{14641}{614656} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{73205 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 3080 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1144640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65856000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{32928 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14641/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 73205/32928*(3*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)))^7 + 3080*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)))^5 + 1144640*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3
- 65856000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^4