3.2480 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{19415 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}+\frac{185 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}-\frac{222185 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(196*(2 + 3*x)^2) + (19415*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (22
2185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.226128, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{19415 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}+\frac{185 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}-\frac{222185 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(196*(2 + 3*x)^2) + (19415*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (22
2185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 22.2467, size = 109, normalized size = 0.89 \[ \frac{19415 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2744 \left (3 x + 2\right )} + \frac{185 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \left (3 x + 2\right )^{3}} - \frac{222185 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19415*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2744*(3*x + 2)) + 185*sqrt(-2*x + 1)*sqrt(5*
x + 3)/(196*(3*x + 2)**2) + sqrt(-2*x + 1)*sqrt(5*x + 3)/(7*(3*x + 2)**3) - 2221
85*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208

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Mathematica [A]  time = 0.0990555, size = 77, normalized size = 0.63 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (19415 x^2+26750 x+9248\right )}{(3 x+2)^3}-222185 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416} \]

Antiderivative was successfully verified.

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

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9248 + 26750*x + 19415*x^2))/(2 + 3*x)^3 - 22
2185*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/38416

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Maple [B]  time = 0.023, size = 202, normalized size = 1.7 \[{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 5998995\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7998660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2446290\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1777480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3370500\,x\sqrt{-10\,{x}^{2}-x+3}+1165248\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5998995*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x^3+11997990*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^2+7998660*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x+2446290*x^2*(-10*x^2-x+3)^(1/2)+1777480*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))+3370500*x*(-10*x^2-x+3)^(1/2)+1165248*(-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 = 1.50298, size = 144, normalized size = 1.18 \[ \frac{222185}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{185 \, \sqrt{-10 \, x^{2} - x + 3}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{19415 \, \sqrt{-10 \, x^{2} - x + 3}}{2744 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

222185/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/7*sqr
t(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 185/196*sqrt(-10*x^2 - x + 3)/
(9*x^2 + 12*x + 4) + 19415/2744*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.223051, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (18 \, \sqrt{7}{\left (19415 \, x^{2} + 26750 \, x + 9248\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 222185 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38416 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38416*sqrt(7)*(18*sqrt(7)*(19415*x^2 + 26750*x + 9248)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 222185*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqr
t(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(1/(2+3*x)**4/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.311302, size = 429, normalized size = 3.52 \[ \frac{44437}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{495 \,{\left (937 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 333760 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 35170240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1372 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

44437/76832*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))) + 495/1372*(937*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 + 333760*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 + 35170240*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