3.2341 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{33 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(3136*(2 + 3*x)^2) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(560*(2 +
3*x)^3) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(5*(2 + 3*x)^5) + (33*Sqrt[1 - 2*x]
*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.264159, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{33 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{(1-2 x)^{3/2} (5 x+3)^{7/2}}{5 (3 x+2)^5}-\frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{560 (3 x+2)^3}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6,x]

[Out]

(-43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (1331*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(3136*(2 + 3*x)^2) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(560*(2 +
3*x)^3) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(5*(2 + 3*x)^5) + (33*Sqrt[1 - 2*x]
*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (483153*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 21.5528, size = 163, normalized size = 0.91 \[ - \frac{3993 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )^{2}} - \frac{121 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{784 \left (3 x + 2\right )^{3}} - \frac{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{280 \left (3 x + 2\right )^{4}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{5 \left (3 x + 2\right )^{5}} + \frac{43923 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43904 \left (3 x + 2\right )} - \frac{483153 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**6,x)

[Out]

-3993*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(21952*(3*x + 2)**2) - 121*(-2*x + 1)**(3/
2)*(5*x + 3)**(3/2)/(784*(3*x + 2)**3) - 33*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(
280*(3*x + 2)**4) + (-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/(5*(3*x + 2)**5) + 43923*
sqrt(-2*x + 1)*sqrt(5*x + 3)/(43904*(3*x + 2)) - 483153*sqrt(7)*atan(sqrt(7)*sqr
t(-2*x + 1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.10962, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (15899035 x^4+46076650 x^3+47906548 x^2+21437032 x+3507552\right )}{(3 x+2)^5}-2415765 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{3073280} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3507552 + 21437032*x + 47906548*x^2 + 46076650
*x^3 + 15899035*x^4))/(2 + 3*x)^5 - 2415765*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[
7 - 14*x]*Sqrt[3 + 5*x])])/3073280

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Maple [B]  time = 0.019, size = 298, normalized size = 1.7 \[{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 587030895\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+670691672\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^6,x)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(587030895*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+1956769650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^4+2609026200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^3+222586490*x^4*(-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^2+645073100*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+6706916
72*x^2*(-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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.52674, size = 306, normalized size = 1.7 \[ \frac{90695}{230496} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1221 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{54417 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{153664 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{738705}{153664} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{483153}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{650859}{307328} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{215303 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{921984 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

90695/230496*(-10*x^2 - x + 3)^(3/2) - 1/35*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 8
10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 33/392*(-10*x^2 - x + 3)^(5/2)/(81*x
^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1221/5488*(-10*x^2 - x + 3)^(5/2)/(27*x^3
+ 54*x^2 + 36*x + 8) + 54417/153664*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) +
738705/153664*sqrt(-10*x^2 - x + 3)*x + 483153/614656*sqrt(7)*arcsin(37/11*x/ab
s(3*x + 2) + 20/11/abs(3*x + 2)) - 650859/307328*sqrt(-10*x^2 - x + 3) + 215303/
921984*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.223267, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (15899035 \, x^{4} + 46076650 \, x^{3} + 47906548 \, x^{2} + 21437032 \, x + 3507552\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2415765 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/3073280*sqrt(7)*(2*sqrt(7)*(15899035*x^4 + 46076650*x^3 + 47906548*x^2 + 21437
032*x + 3507552)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 2415765*(243*x^5 + 810*x^4 + 108
0*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.501264, size = 594, normalized size = 3.3 \[ \frac{483153}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{161051 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2007040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 307328000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 18439680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{21952 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="giac")

[Out]

483153/6146560*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)))) - 161051/21952*(3*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 3920*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 + 2007040*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 - 307328000*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 - 18439680000*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-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)^5