3.2277 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^2}-\frac{6401 \sqrt{1-2 x} \sqrt{5 x+3}}{10584 (3 x+2)}-\frac{50}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{250433 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{31752 \sqrt{7}} \]

[Out]

(-6401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10584*(2 + 3*x)) - (59*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(252*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) -
(50*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (250433*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(31752*Sqrt[7])

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Rubi [A]  time = 0.312666, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^2}-\frac{6401 \sqrt{1-2 x} \sqrt{5 x+3}}{10584 (3 x+2)}-\frac{50}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{250433 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{31752 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]

[Out]

(-6401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10584*(2 + 3*x)) - (59*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(252*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) -
(50*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (250433*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/(31752*Sqrt[7])

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Rubi in Sympy [A]  time = 29.6264, size = 136, normalized size = 0.91 \[ - \frac{6401 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10584 \left (3 x + 2\right )} - \frac{59 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{252 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{3}} - \frac{50 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{81} - \frac{250433 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{222264} \]

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)**4,x)

[Out]

-6401*sqrt(-2*x + 1)*sqrt(5*x + 3)/(10584*(3*x + 2)) - 59*sqrt(-2*x + 1)*(5*x +
3)**(3/2)/(252*(3*x + 2)**2) - sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(9*(3*x + 2)**3)
- 50*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/81 - 250433*sqrt(7)*atan(sqrt(7)*s
qrt(-2*x + 1)/(7*sqrt(5*x + 3)))/222264

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Mathematica [A]  time = 0.183951, size = 112, normalized size = 0.75 \[ \frac{-\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (124179 x^2+159174 x+51056\right )}{(3 x+2)^3}-250433 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-137200 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{444528} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^4,x]

[Out]

((-42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(51056 + 159174*x + 124179*x^2))/(2 + 3*x)^3 -
250433*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 137200*S
qrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/444528

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Maple [B]  time = 0.019, size = 253, normalized size = 1.7 \[{\frac{1}{444528\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6761691\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-3704400\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-7408800\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+9015588\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-4939200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-5215518\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2003464\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1097600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -6685308\,x\sqrt{-10\,{x}^{2}-x+3}-2144352\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^4,x)

[Out]

1/444528*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6761691*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^3-3704400*10^(1/2)*arcsin(20/11*x+1/11)*x^3+13523382*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-7408800*10^(1/2)*
arcsin(20/11*x+1/11)*x^2+9015588*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x-4939200*10^(1/2)*arcsin(20/11*x+1/11)*x-5215518*x^2*(-10*x^2-x+3)^
(1/2)+2003464*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1097600
*10^(1/2)*arcsin(20/11*x+1/11)-6685308*x*(-10*x^2-x+3)^(1/2)-2144352*(-10*x^2-x+
3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.56994, size = 178, normalized size = 1.19 \[ -\frac{25}{81} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{250433}{444528} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{515}{2646} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{63 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{103 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{588 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{5989 \, \sqrt{-10 \, x^{2} - x + 3}}{10584 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

-25/81*sqrt(10)*arcsin(20/11*x + 1/11) + 250433/444528*sqrt(7)*arcsin(37/11*x/ab
s(3*x + 2) + 20/11/abs(3*x + 2)) - 515/2646*sqrt(-10*x^2 - x + 3) + 1/63*(-10*x^
2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) - 103/588*(-10*x^2 - x + 3)^(3/2)/
(9*x^2 + 12*x + 4) - 5989/10584*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.233752, size = 192, normalized size = 1.29 \[ -\frac{\sqrt{7}{\left (19600 \, \sqrt{10} \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (124179 \, x^{2} + 159174 \, x + 51056\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 250433 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{444528 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

-1/444528*sqrt(7)*(19600*sqrt(10)*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/
20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(124179*x^2 +
159174*x + 51056)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 250433*(27*x^3 + 54*x^2 + 36*x
+ 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(27*x^3 +
54*x^2 + 36*x + 8)

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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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.398965, size = 520, normalized size = 3.49 \[ \frac{250433}{4445280} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{25}{81} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (6401 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 4674880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1034801600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{5292 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="giac")

[Out]

250433/4445280*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)))) - 25/81*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1
1/5292*(6401*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 4674880*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 + 1034801600*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)^3