3.2600 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{3471145 \sqrt{5 x+3}}{3486252 \sqrt{1-2 x}}+\frac{423 \sqrt{5 x+3}}{56 (1-2 x)^{3/2} (3 x+2)}-\frac{101485 \sqrt{5 x+3}}{45276 (1-2 x)^{3/2}}+\frac{193 \sqrt{5 x+3}}{196 (1-2 x)^{3/2} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}-\frac{330255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(-101485*Sqrt[3 + 5*x])/(45276*(1 - 2*x)^(3/2)) - (3471145*Sqrt[3 + 5*x])/(34862
52*Sqrt[1 - 2*x]) + Sqrt[3 + 5*x]/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (193*Sqrt[3
+ 5*x])/(196*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (423*Sqrt[3 + 5*x])/(56*(1 - 2*x)^(3
/2)*(2 + 3*x)) - (330255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*S
qrt[7])

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Rubi [A]  time = 0.403448, antiderivative size = 166, 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 \[ -\frac{3471145 \sqrt{5 x+3}}{3486252 \sqrt{1-2 x}}+\frac{423 \sqrt{5 x+3}}{56 (1-2 x)^{3/2} (3 x+2)}-\frac{101485 \sqrt{5 x+3}}{45276 (1-2 x)^{3/2}}+\frac{193 \sqrt{5 x+3}}{196 (1-2 x)^{3/2} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}-\frac{330255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-101485*Sqrt[3 + 5*x])/(45276*(1 - 2*x)^(3/2)) - (3471145*Sqrt[3 + 5*x])/(34862
52*Sqrt[1 - 2*x]) + Sqrt[3 + 5*x]/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (193*Sqrt[3
+ 5*x])/(196*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (423*Sqrt[3 + 5*x])/(56*(1 - 2*x)^(3
/2)*(2 + 3*x)) - (330255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*S
qrt[7])

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Rubi in Sympy [A]  time = 36.1777, size = 151, normalized size = 0.91 \[ - \frac{330255 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{134456} - \frac{3471145 \sqrt{5 x + 3}}{3486252 \sqrt{- 2 x + 1}} - \frac{101485 \sqrt{5 x + 3}}{45276 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{423 \sqrt{5 x + 3}}{56 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{193 \sqrt{5 x + 3}}{196 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} + \frac{\sqrt{5 x + 3}}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-330255*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/134456 - 3471145*
sqrt(5*x + 3)/(3486252*sqrt(-2*x + 1)) - 101485*sqrt(5*x + 3)/(45276*(-2*x + 1)*
*(3/2)) + 423*sqrt(5*x + 3)/(56*(-2*x + 1)**(3/2)*(3*x + 2)) + 193*sqrt(5*x + 3)
/(196*(-2*x + 1)**(3/2)*(3*x + 2)**2) + sqrt(5*x + 3)/(7*(-2*x + 1)**(3/2)*(3*x
+ 2)**3)

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Mathematica [A]  time = 0.127635, size = 87, normalized size = 0.52 \[ \frac{\sqrt{5 x+3} \left (374883660 x^4+140350860 x^3-244982277 x^2-48873610 x+44829024\right )}{6972504 (1-2 x)^{3/2} (3 x+2)^3}-\frac{330255 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x]*(44829024 - 48873610*x - 244982277*x^2 + 140350860*x^3 + 37488366
0*x^4))/(6972504*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (330255*ArcTan[(-20 - 37*x)/(2*S
qrt[7 - 14*x]*Sqrt[3 + 5*x])])/(38416*Sqrt[7])

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Maple [B]  time = 0.024, size = 305, normalized size = 1.8 \[{\frac{1}{97615056\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) ^{2}} \left ( 12947317020\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-5394715425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6953188770\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-3429751878\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -684230540\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/97615056*(12947317020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^5+12947317020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4-5394715425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+5248
371240*x^4*(-10*x^2-x+3)^(1/2)-6953188770*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^2+1964912040*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-3429751878*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))-68423
0540*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^
(1/2)/(2+3*x)^3/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{4}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(5/2)), x)

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Fricas [A]  time = 0.224093, size = 167, normalized size = 1.01 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (374883660 \, x^{4} + 140350860 \, x^{3} - 244982277 \, x^{2} - 48873610 \, x + 44829024\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 119882565 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{97615056 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/97615056*sqrt(7)*(2*sqrt(7)*(374883660*x^4 + 140350860*x^3 - 244982277*x^2 - 4
8873610*x + 44829024)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 119882565*(108*x^5 + 108*x^
4 - 45*x^3 - 58*x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*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(1/(1-2*x)**(5/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.496346, size = 482, normalized size = 2.9 \[ \frac{66051}{537824} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \,{\left (932 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5511 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{152523525 \,{\left (2 \, x - 1\right )}^{2}} + \frac{297 \,{\left (15599 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 5723200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 607208000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{67228 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

66051/537824*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 32/152523525*(932*sqrt(5)*(5*x + 3) - 5511*sqrt(5))*sqrt(5*x + 3)*sq
rt(-10*x + 5)/(2*x - 1)^2 + 297/67228*(15599*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 + 5723200*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)))^3 + 607208000*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*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