3.2268 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{152571047 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)}+\frac{1460201 \sqrt{1-2 x} \sqrt{5 x+3}}{2370816 (3 x+2)^2}+\frac{42461 \sqrt{1-2 x} \sqrt{5 x+3}}{423360 (3 x+2)^3}+\frac{4619 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{3780 (3 x+2)^5}-\frac{64645339 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3780*(2 + 3*x)^5) + (4619*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(211680*(2 + 3*x)^4) + (42461*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(423360*(2
+ 3*x)^3) + (1460201*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2370816*(2 + 3*x)^2) + (1525
71047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(33191424*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*
x)^(3/2))/(18*(2 + 3*x)^6) - (64645339*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1229312*Sqrt[7])

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Rubi [A]  time = 0.439464, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{152571047 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)}+\frac{1460201 \sqrt{1-2 x} \sqrt{5 x+3}}{2370816 (3 x+2)^2}+\frac{42461 \sqrt{1-2 x} \sqrt{5 x+3}}{423360 (3 x+2)^3}+\frac{4619 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{3780 (3 x+2)^5}-\frac{64645339 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^7,x]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3780*(2 + 3*x)^5) + (4619*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(211680*(2 + 3*x)^4) + (42461*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(423360*(2
+ 3*x)^3) + (1460201*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2370816*(2 + 3*x)^2) + (1525
71047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(33191424*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*
x)^(3/2))/(18*(2 + 3*x)^6) - (64645339*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1229312*Sqrt[7])

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Rubi in Sympy [A]  time = 43.6324, size = 190, normalized size = 0.91 \[ \frac{152571047 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{33191424 \left (3 x + 2\right )} + \frac{1460201 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2370816 \left (3 x + 2\right )^{2}} + \frac{42461 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{423360 \left (3 x + 2\right )^{3}} + \frac{4619 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{211680 \left (3 x + 2\right )^{4}} - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3780 \left (3 x + 2\right )^{5}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{18 \left (3 x + 2\right )^{6}} - \frac{64645339 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{8605184} \]

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)**7,x)

[Out]

152571047*sqrt(-2*x + 1)*sqrt(5*x + 3)/(33191424*(3*x + 2)) + 1460201*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(2370816*(3*x + 2)**2) + 42461*sqrt(-2*x + 1)*sqrt(5*x + 3)/(
423360*(3*x + 2)**3) + 4619*sqrt(-2*x + 1)*sqrt(5*x + 3)/(211680*(3*x + 2)**4) -
107*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3780*(3*x + 2)**5) - sqrt(-2*x + 1)*(5*x + 3)
**(3/2)/(18*(3*x + 2)**6) - 64645339*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt
(5*x + 3)))/8605184

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Mathematica [A]  time = 0.14822, size = 92, normalized size = 0.44 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (20597091345 x^5+69576897780 x^4+94045700016 x^3+63585046048 x^2+21497808880 x+2906375616\right )}{(3 x+2)^6}-8727120765 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2323399680} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^7,x]

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2906375616 + 21497808880*x + 63585046048*x^2
+ 94045700016*x^3 + 69576897780*x^4 + 20597091345*x^5))/(2 + 3*x)^6 - 8727120765
*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/2323399680

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Maple [B]  time = 0.019, size = 346, normalized size = 1.7 \[{\frac{1}{258155520\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 706896781965\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+890190644672\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^7,x)

[Out]

1/258155520*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(706896781965*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+2827587127860*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+4712645213100*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))*x^4+288359278830*x^5*(-10*x^2-x+3)^(1/2)[phone]
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+974076568920*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+1316639800224*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+890190644672*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))+3
00969324320*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)^6

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Maxima [A]  time = 1.55462, size = 329, normalized size = 1.57 \[ \frac{64645339}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2671295}{921984} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{42 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{29 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{980 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{1273 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{7840 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{45245 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{65856 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1602777 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{19767583 \, \sqrt{-10 \, x^{2} - x + 3}}{3687936 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

64645339/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 26
71295/921984*sqrt(-10*x^2 - x + 3) - 1/42*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 291
6*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 29/980*(-10*x^2 - x + 3)^
(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 1273/7840*(-10*x^2
- x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 45245/65856*(-10*x^2
- x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 1602777/614656*(-10*x^2 - x + 3)^(
3/2)/(9*x^2 + 12*x + 4) - 19767583/3687936*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.225185, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (20597091345 \, x^{5} + 69576897780 \, x^{4} + 94045700016 \, x^{3} + 63585046048 \, x^{2} + 21497808880 \, x + 2906375616\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 969680085 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{258155520 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/258155520*sqrt(7)*(2*sqrt(7)*(20597091345*x^5 + 69576897780*x^4 + 94045700016*
x^3 + 63585046048*x^2 + 21497808880*x + 2906375616)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 969680085*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*
arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*x^6 + 2916
*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.548883, size = 676, normalized size = 3.23 \[ \frac{64645339}{172103680} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (145707 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 231188440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 144245619840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 41365512115200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 5067855403520000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 250767109017600000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1843968 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="giac")

[Out]

64645339/172103680*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)))) - 1331/1843968*(145707*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)))^11 +
231188440*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)))^9 - 144245619840*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^7 - 41365512115200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5
- 5067855403520000*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 - 250767109017600000*s
qrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s
qrt(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)^6