3.2266 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{115 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}-\frac{1265 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{13915 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{153065 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(-13915*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (1265*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(4704*(2 + 3*x)^2) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(28*(2 +
3*x)^4) + (115*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*x)^3) - (153065*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.214296, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{115 \sqrt{1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}-\frac{1265 \sqrt{1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac{13915 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}-\frac{153065 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-13915*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (1265*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(4704*(2 + 3*x)^2) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(28*(2 +
3*x)^4) + (115*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(168*(2 + 3*x)^3) - (153065*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 16.6636, size = 138, normalized size = 0.91 \[ - \frac{1265 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{10976 \left (3 x + 2\right )^{2}} - \frac{115 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1176 \left (3 x + 2\right )^{3}} + \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{28 \left (3 x + 2\right )^{4}} + \frac{13915 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )} - \frac{153065 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1265*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(10976*(3*x + 2)**2) - 115*(-2*x + 1)**(3/
2)*(5*x + 3)**(3/2)/(1176*(3*x + 2)**3) + 3*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(
28*(3*x + 2)**4) + 13915*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21952*(3*x + 2)) - 153065
*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.107597, size = 82, normalized size = 0.54 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (1104135 x^3+2269240 x^2+1512052 x+328464\right )}{(3 x+2)^4}-4132755 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{8297856} \]

Antiderivative was successfully verified.

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

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(328464 + 1512052*x + 2269240*x^2 + 1104135*x^
3))/(2 + 3*x)^4 - 4132755*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 +
5*x])])/8297856

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Maple [B]  time = 0.017, size = 250, normalized size = 1.7 \[{\frac{1}{921984\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 37194795\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+31769360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7347120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+4598496\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/921984*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37194795*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+99186120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^3+99186120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^2+15457890*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+31769360*x^2*(-10*x^2-x+3)^(1/2)+7347120*7^
(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3
)^(1/2)+4598496*(-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 = 1.51442, size = 212, normalized size = 1.4 \[ \frac{153065}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{6325}{16464} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{95 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{3795 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{46805 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153065/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 6325/1
6464*sqrt(-10*x^2 - x + 3) - 1/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 21
6*x^2 + 96*x + 16) + 95/1176*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8
) + 3795/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 46805/65856*sqrt(-10
*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.222446, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1104135 \, x^{3} + 2269240 \, x^{2} + 1512052 \, x + 328464\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 459195 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{921984 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/921984*sqrt(7)*(2*sqrt(7)*(1104135*x^3 + 2269240*x^2 + 1512052*x + 328464)*sqr
t(5*x + 3)*sqrt(-2*x + 1) + 459195*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arct
an(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 +
216*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)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.404095, size = 512, normalized size = 3.39 \[ \frac{30613}{614656} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{6655 \,{\left (69 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 70840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 15821120 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 1514688000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{32928 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30613/614656*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)))) - 6655/32928*(69*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 + 70840*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)))^5 - 15821120*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)))
^3 - 1514688000*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)^4