3.2543 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]

[Out]

(-7986105*Sqrt[3 + 5*x])/(845152*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/(28*Sqrt[1 -
2*x]*(2 + 3*x)^4) + (263*Sqrt[3 + 5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (6621
*Sqrt[3 + 5*x])/(1568*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (698295*Sqrt[3 + 5*x])/(21952
*Sqrt[1 - 2*x]*(2 + 3*x)) - (24922335*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(153664*Sqrt[7])

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Rubi [A]  time = 0.412221, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-7986105*Sqrt[3 + 5*x])/(845152*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/(28*Sqrt[1 -
2*x]*(2 + 3*x)^4) + (263*Sqrt[3 + 5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (6621
*Sqrt[3 + 5*x])/(1568*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (698295*Sqrt[3 + 5*x])/(21952
*Sqrt[1 - 2*x]*(2 + 3*x)) - (24922335*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(153664*Sqrt[7])

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Rubi in Sympy [A]  time = 36.9032, size = 160, normalized size = 0.92 \[ - \frac{24922335 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1075648} - \frac{7986105 \sqrt{5 x + 3}}{845152 \sqrt{- 2 x + 1}} + \frac{698295 \sqrt{5 x + 3}}{21952 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{6621 \sqrt{5 x + 3}}{1568 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{263 \sqrt{5 x + 3}}{392 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{3 \sqrt{5 x + 3}}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24922335*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/1075648 - 79861
05*sqrt(5*x + 3)/(845152*sqrt(-2*x + 1)) + 698295*sqrt(5*x + 3)/(21952*sqrt(-2*x
+ 1)*(3*x + 2)) + 6621*sqrt(5*x + 3)/(1568*sqrt(-2*x + 1)*(3*x + 2)**2) + 263*s
qrt(5*x + 3)/(392*sqrt(-2*x + 1)*(3*x + 2)**3) + 3*sqrt(5*x + 3)/(28*sqrt(-2*x +
1)*(3*x + 2)**4)

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Mathematica [A]  time = 0.143821, size = 87, normalized size = 0.5 \[ \frac{-\frac{14 \sqrt{5 x+3} \left (1293749010 x^4+1998242055 x^3+482249808 x^2-491393004 x-205593328\right )}{\sqrt{1-2 x} (3 x+2)^4}-274145685 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{23664256} \]

Antiderivative was successfully verified.

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

[Out]

((-14*Sqrt[3 + 5*x]*(-205593328 - 491393004*x + 482249808*x^2 + 1998242055*x^3 +
1293749010*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^4) - 274145685*Sqrt[7]*ArcTan[(-20 -
37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/23664256

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Maple [B]  time = 0.027, size = 305, normalized size = 1.8 \[{\frac{1}{23664256\, \left ( 2+3\,x \right ) ^{4} \left ( -1+2\,x \right ) } \left ( 44411600970\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6579496440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17545323840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6751497312\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4386330960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6879502056\,x\sqrt{-10\,{x}^{2}-x+3}-2878306592\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23664256*(44411600970*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^5+96225135435*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+59215467960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+181
12486140*x^4*(-10*x^2-x+3)^(1/2)-6579496440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^2+27975388770*x^3*(-10*x^2-x+3)^(1/2)-17545323840*7^(1/
2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+6751497312*x^2*(-10*x^2-
x+3)^(1/2)-4386330960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
-6879502056*x*(-10*x^2-x+3)^(1/2)-2878306592*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*
(1-2*x)^(1/2)/(2+3*x)^4/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{5}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^5*(-2*x + 1)^(3/2)), x)

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Fricas [A]  time = 0.237871, size = 167, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1293749010 \, x^{4} + 1998242055 \, x^{3} + 482249808 \, x^{2} - 491393004 \, x - 205593328\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 274145685 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{23664256 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23664256*sqrt(7)*(2*sqrt(7)*(1293749010*x^4 + 1998242055*x^3 + 482249808*x^2 -
491393004*x - 205593328)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 274145685*(162*x^5 + 35
1*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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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(1/(1-2*x)**(3/2)/(2+3*x)**5/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.569646, size = 547, normalized size = 3.16 \[ \frac{4984467}{4302592} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{924385 \,{\left (2 \, x - 1\right )}} + \frac{99 \,{\left (4411181 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2388710520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 506212728000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 38676680000000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{537824 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4984467/4302592*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)))) - 64/924385*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 99/53
7824*(4411181*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 2388710520*sqrt(10)*((sqr
t(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 + 506212728000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 +
38676680000000*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)^4