3.1831 \(\int \frac{\sqrt{1-2 x} (2+3 x)^4}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{10 (5 x+3)^2}-\frac{131 \sqrt{1-2 x} (3 x+2)^3}{550 (5 x+3)}+\frac{1428 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}-\frac{12803 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

[Out]

(-21*(704 - 375*x)*Sqrt[1 - 2*x])/68750 + (1428*Sqrt[1 - 2*x]*(2 + 3*x)^2)/6875
- (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(10*(3 + 5*x)^2) - (131*Sqrt[1 - 2*x]*(2 + 3*x)^3)
/(550*(3 + 5*x)) - (12803*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55])

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Rubi [A]  time = 0.20493, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{10 (5 x+3)^2}-\frac{131 \sqrt{1-2 x} (3 x+2)^3}{550 (5 x+3)}+\frac{1428 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}-\frac{12803 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-21*(704 - 375*x)*Sqrt[1 - 2*x])/68750 + (1428*Sqrt[1 - 2*x]*(2 + 3*x)^2)/6875
- (Sqrt[1 - 2*x]*(2 + 3*x)^4)/(10*(3 + 5*x)^2) - (131*Sqrt[1 - 2*x]*(2 + 3*x)^3)
/(550*(3 + 5*x)) - (12803*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55])

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Rubi in Sympy [A]  time = 26.3079, size = 104, normalized size = 0.87 \[ - \frac{\left (- 118125 x + 221760\right ) \sqrt{- 2 x + 1}}{1031250} - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{10 \left (5 x + 3\right )^{2}} - \frac{131 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{550 \left (5 x + 3\right )} + \frac{1428 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{6875} - \frac{12803 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1890625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-118125*x + 221760)*sqrt(-2*x + 1)/1031250 - sqrt(-2*x + 1)*(3*x + 2)**4/(10*(
5*x + 3)**2) - 131*sqrt(-2*x + 1)*(3*x + 2)**3/(550*(5*x + 3)) + 1428*sqrt(-2*x
+ 1)*(3*x + 2)**2/6875 - 12803*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/189062
5

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Mathematica [A]  time = 0.133484, size = 68, normalized size = 0.57 \[ \frac{\frac{55 \sqrt{1-2 x} \left (445500 x^4+1103850 x^3+506880 x^2-200305 x-121976\right )}{(5 x+3)^2}-25606 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3781250} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-121976 - 200305*x + 506880*x^2 + 1103850*x^3 + 445500*x^4))
/(3 + 5*x)^2 - 25606*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3781250

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[{\frac{81}{1250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{369}{1250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{108}{3125}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{263}{220} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{53}{20}\sqrt{1-2\,x}} \right ) }-{\frac{12803\,\sqrt{55}}{1890625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/1250*(1-2*x)^(5/2)-369/1250*(1-2*x)^(3/2)+108/3125*(1-2*x)^(1/2)+4/125*(263/2
20*(1-2*x)^(3/2)-53/20*(1-2*x)^(1/2))/(-6-10*x)^2-12803/1890625*arctanh(1/11*55^
(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48311, size = 136, normalized size = 1.13 \[ \frac{81}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{369}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12803}{3781250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{108}{3125} \, \sqrt{-2 \, x + 1} + \frac{263 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 583 \, \sqrt{-2 \, x + 1}}{6875 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/1250*(-2*x + 1)^(5/2) - 369/1250*(-2*x + 1)^(3/2) + 12803/3781250*sqrt(55)*lo
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 108/3125*sqrt(
-2*x + 1) + 1/6875*(263*(-2*x + 1)^(3/2) - 583*sqrt(-2*x + 1))/(25*(2*x - 1)^2 +
220*x + 11)

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Fricas [A]  time = 0.214887, size = 120, normalized size = 1. \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (445500 \, x^{4} + 1103850 \, x^{3} + 506880 \, x^{2} - 200305 \, x - 121976\right )} \sqrt{-2 \, x + 1} + 12803 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{3781250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3781250*sqrt(55)*(sqrt(55)*(445500*x^4 + 1103850*x^3 + 506880*x^2 - 200305*x -
121976)*sqrt(-2*x + 1) + 12803*(25*x^2 + 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55
*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213187, size = 138, normalized size = 1.15 \[ \frac{81}{1250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{369}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12803}{3781250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{108}{3125} \, \sqrt{-2 \, x + 1} + \frac{263 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 583 \, \sqrt{-2 \, x + 1}}{27500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/1250*(2*x - 1)^2*sqrt(-2*x + 1) - 369/1250*(-2*x + 1)^(3/2) + 12803/3781250*s
qrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 108/3125*sqrt(-2*x + 1) + 1/27500*(263*(-2*x + 1)^(3/2) - 583*sqrt(-2*x + 1
))/(5*x + 3)^2