3.2279 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{17 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac{187 \sqrt{1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac{2057 \sqrt{1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac{22627 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{248897 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-22627*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (2057*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(9408*(2 + 3*x)^2) - (187*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1680*(2
+ 3*x)^3) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(35*(2 + 3*x)^5) + (17*Sqrt[1 -
2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (248897*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.269122, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{17 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac{187 \sqrt{1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac{2057 \sqrt{1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac{22627 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{248897 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-22627*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (2057*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(9408*(2 + 3*x)^2) - (187*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1680*(2
+ 3*x)^3) + (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(35*(2 + 3*x)^5) + (17*Sqrt[1 -
2*x]*(3 + 5*x)^(7/2))/(40*(2 + 3*x)^4) - (248897*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 20.5952, size = 165, normalized size = 0.92 \[ - \frac{187 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{2352 \left (3 x + 2\right )^{3}} - \frac{17 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{280 \left (3 x + 2\right )^{4}} + \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{35 \left (3 x + 2\right )^{5}} - \frac{22627 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43904 \left (3 x + 2\right )} + \frac{2057 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3136 \left (3 x + 2\right )^{2}} - \frac{248897 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-187*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(2352*(3*x + 2)**3) - 17*(-2*x + 1)**(3/
2)*(5*x + 3)**(5/2)/(280*(3*x + 2)**4) + 3*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/(3
5*(3*x + 2)**5) - 22627*sqrt(-2*x + 1)*sqrt(5*x + 3)/(43904*(3*x + 2)) + 2057*sq
rt(-2*x + 1)*(5*x + 3)**(3/2)/(3136*(3*x + 2)**2) - 248897*sqrt(7)*atan(sqrt(7)*
sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.150064, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (27422145 x^4+74915550 x^3+74550556 x^2+32206264 x+5112864\right )}{(3 x+2)^5}-3733455 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{9219840} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5112864 + 32206264*x + 74550556*x^2 + 74915550
*x^3 + 27422145*x^4))/(2 + 3*x)^5 - 3733455*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[
7 - 14*x]*Sqrt[3 + 5*x])])/9219840

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Maple [B]  time = 0.017, size = 298, normalized size = 1.7 \[{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 907229565\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1043707784\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(907229565*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+3024098550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^4+4032131400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^3+383910030*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+1048817700*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+104370
7784*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/
(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.51486, size = 267, normalized size = 1.48 \[ \frac{248897}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{10285}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{105 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{40 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{6171 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{76109 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

248897/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 10285/
32928*sqrt(-10*x^2 - x + 3) + 1/105*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32) - 3/40*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x
^3 + 216*x^2 + 96*x + 16) + 45/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36
*x + 8) + 6171/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 76109/131712*s
qrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.22268, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (27422145 \, x^{4} + 74915550 \, x^{3} + 74550556 \, x^{2} + 32206264 \, x + 5112864\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3733455 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{9219840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*sqrt(7)*(2*sqrt(7)*(27422145*x^4 + 74915550*x^3 + 74550556*x^2 + 32206
264*x + 5112864)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3733455*(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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.485157, size = 594, normalized size = 3.3 \[ \frac{248897}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (51 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 66640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 34119680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 3618944000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 313474560000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{65856 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

248897/6146560*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)))) - 14641/65856*(51*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 + 66640*sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 34119680*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 - 3618944000*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 - 313474560000*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)^5