3.2468 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}-\frac{55 \sqrt{1-2 x} (5 x+3)^{3/2}}{588 (3 x+2)^2}-\frac{605 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (55*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(588*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^3) -
(6655*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.172459, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}-\frac{55 \sqrt{1-2 x} (5 x+3)^{3/2}}{588 (3 x+2)^2}-\frac{605 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (55*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(588*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^3) -
(6655*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 13.6123, size = 110, normalized size = 0.9 \[ - \frac{605 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2744 \left (3 x + 2\right )} - \frac{55 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{588 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{21 \left (3 x + 2\right )^{3}} - \frac{6655 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-605*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2744*(3*x + 2)) - 55*sqrt(-2*x + 1)*(5*x + 3)
**(3/2)/(588*(3*x + 2)**2) - sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(21*(3*x + 2)**3) -
6655*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208

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Mathematica [A]  time = 0.0856438, size = 77, normalized size = 0.63 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (37685 x^2+48170 x+15408\right )}{8232 (3 x+2)^3}-\frac{6655 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{5488 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(15408 + 48170*x + 37685*x^2))/(8232*(2 + 3*x)^3)
- (6655*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(5488*Sqrt[7])

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Maple [B]  time = 0.019, size = 202, normalized size = 1.7 \[{\frac{1}{115248\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 539055\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1078110\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+718740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-527590\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+159720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -674380\,x\sqrt{-10\,{x}^{2}-x+3}-215712\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/115248*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(539055*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x^3+1078110*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^2+718740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))*x-527590*x^2*(-10*x^2-x+3)^(1/2)+159720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))-674380*x*(-10*x^2-x+3)^(1/2)-215712*(-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 = 1.50089, size = 144, normalized size = 1.18 \[ \frac{6655}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{189 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{445 \, \sqrt{-10 \, x^{2} - x + 3}}{5292 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{37685 \, \sqrt{-10 \, x^{2} - x + 3}}{74088 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6655/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/189*sqr
t(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 445/5292*sqrt(-10*x^2 - x + 3)
/(9*x^2 + 12*x + 4) - 37685/74088*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.222241, size = 127, normalized size = 1.04 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (37685 \, x^{2} + 48170 \, x + 15408\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 19965 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{115248 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/115248*sqrt(7)*(2*sqrt(7)*(37685*x^2 + 48170*x + 15408)*sqrt(5*x + 3)*sqrt(-2
*x + 1) - 19965*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.364333, size = 429, normalized size = 3.52 \[ \frac{1331}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{6655 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 2240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 517440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4116 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/76832*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)))) - 6655/4116*(3*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 + 2240*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)))^3 + 517440*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sq
rt(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