3.2280 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1260 (3 x+2)^5}+\frac{106751933 \sqrt{1-2 x} \sqrt{5 x+3}}{99574272 (3 x+2)}+\frac{1057139 \sqrt{1-2 x} \sqrt{5 x+3}}{7112448 (3 x+2)^2}+\frac{47279 \sqrt{1-2 x} \sqrt{5 x+3}}{1270080 (3 x+2)^3}-\frac{6533 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{15036307 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

[Out]

(-6533*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(211680*(2 + 3*x)^4) + (47279*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(1270080*(2 + 3*x)^3) + (1057139*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(71
12448*(2 + 3*x)^2) + (106751933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(99574272*(2 + 3*x)
) - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1260*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 +
5*x)^(5/2))/(18*(2 + 3*x)^6) - (15036307*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(1229312*Sqrt[7])

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Rubi [A]  time = 0.453585, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1260 (3 x+2)^5}+\frac{106751933 \sqrt{1-2 x} \sqrt{5 x+3}}{99574272 (3 x+2)}+\frac{1057139 \sqrt{1-2 x} \sqrt{5 x+3}}{7112448 (3 x+2)^2}+\frac{47279 \sqrt{1-2 x} \sqrt{5 x+3}}{1270080 (3 x+2)^3}-\frac{6533 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{15036307 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-6533*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(211680*(2 + 3*x)^4) + (47279*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(1270080*(2 + 3*x)^3) + (1057139*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(71
12448*(2 + 3*x)^2) + (106751933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(99574272*(2 + 3*x)
) - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1260*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 +
5*x)^(5/2))/(18*(2 + 3*x)^6) - (15036307*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(1229312*Sqrt[7])

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Rubi in Sympy [A]  time = 43.8665, size = 190, normalized size = 0.91 \[ \frac{106751933 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{99574272 \left (3 x + 2\right )} + \frac{1057139 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7112448 \left (3 x + 2\right )^{2}} + \frac{47279 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1270080 \left (3 x + 2\right )^{3}} - \frac{6533 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{211680 \left (3 x + 2\right )^{4}} - \frac{59 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{1260 \left (3 x + 2\right )^{5}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{18 \left (3 x + 2\right )^{6}} - \frac{15036307 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{8605184} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

106751933*sqrt(-2*x + 1)*sqrt(5*x + 3)/(99574272*(3*x + 2)) + 1057139*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(7112448*(3*x + 2)**2) + 47279*sqrt(-2*x + 1)*sqrt(5*x + 3)/(
1270080*(3*x + 2)**3) - 6533*sqrt(-2*x + 1)*sqrt(5*x + 3)/(211680*(3*x + 2)**4)
- 59*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(1260*(3*x + 2)**5) - sqrt(-2*x + 1)*(5*x +
3)**(5/2)/(18*(3*x + 2)**6) - 15036307*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*s
qrt(5*x + 3)))/8605184

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Mathematica [A]  time = 0.132509, size = 92, normalized size = 0.44 \[ \frac{\frac{378 \sqrt{1-2 x} \sqrt{5 x+3} \left (4803836985 x^5+16234789140 x^4+21960917808 x^3+14818971424 x^2+4978384240 x+665270208\right )}{(3 x+2)^6}-6089704335 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{6970199040} \]

Antiderivative was successfully verified.

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

[Out]

((378*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(665270208 + 4978384240*x + 14818971424*x^2 +
21960917808*x^3 + 16234789140*x^4 + 4803836985*x^5))/(2 + 3*x)^6 - 6089704335*Sq
rt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/6970199040

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Maple [B]  time = 0.02, size = 346, normalized size = 1.7 \[{\frac{1}{258155520\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 164422017045\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+207465599936\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^7,x)

[Out]

1/258155520*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(164422017045*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+657688068180*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+1096146780300*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^4+67253717790*x^5*(-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^3+227287047960*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+307452849312*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+207465599936*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))+696973
79360*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 = 1.505, size = 329, normalized size = 1.57 \[ \frac{15036307}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{621335}{921984} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{126 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2940 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{547 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{23520 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{31055 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{197568 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{372801 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{4597879 \, \sqrt{-10 \, x^{2} - x + 3}}{3687936 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15036307/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 62
1335/921984*sqrt(-10*x^2 - x + 3) + 1/126*(-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) - 169/2940*(-10*x^2 - x + 3
)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 547/23520*(-10*x
^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 31055/197568*(-10*x
^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 372801/614656*(-10*x^2 - x + 3)
^(3/2)/(9*x^2 + 12*x + 4) - 4597879/3687936*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.223307, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (4803836985 \, x^{5} + 16234789140 \, x^{4} + 21960917808 \, x^{3} + 14818971424 \, x^{2} + 4978384240 \, x + 665270208\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 225544605 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{258155520 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/258155520*sqrt(7)*(2*sqrt(7)*(4803836985*x^5 + 16234789140*x^4 + 21960917808*x
^3 + 14818971424*x^2 + 4978384240*x + 665270208)*sqrt(5*x + 3)*sqrt(-2*x + 1) +
225544605*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arc
tan(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)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.600638, size = 676, normalized size = 3.23 \[ \frac{15036307}{172103680} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (3081 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 4888520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3188465280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 599903001600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 103716175360000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 5302514380800000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1843968 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15036307/172103680*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)))) - 14641/1843968*(3081*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 +
4888520*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 + 3188465280*sqrt(10)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^7 - 599903001600*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 - 10371
6175360000*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 - 5302514380800000*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)^6