3.2181 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=152 \[ -\frac{7554245}{5021863 \sqrt{1-2 x}}+\frac{32765}{1694 (1-2 x)^{3/2} (5 x+3)}-\frac{667615}{195657 (1-2 x)^{3/2}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac{505}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac{17820}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{738625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

-667615/(195657*(1 - 2*x)^(3/2)) - 7554245/(5021863*Sqrt[1 - 2*x]) - 505/(154*(1
- 2*x)^(3/2)*(3 + 5*x)^2) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 32765
/(1694*(1 - 2*x)^(3/2)*(3 + 5*x)) + (17820*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.419194, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{7554245}{5021863 \sqrt{1-2 x}}+\frac{32765}{1694 (1-2 x)^{3/2} (5 x+3)}-\frac{667615}{195657 (1-2 x)^{3/2}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac{505}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac{17820}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{738625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

-667615/(195657*(1 - 2*x)^(3/2)) - 7554245/(5021863*Sqrt[1 - 2*x]) - 505/(154*(1
- 2*x)^(3/2)*(3 + 5*x)^2) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 32765
/(1694*(1 - 2*x)^(3/2)*(3 + 5*x)) + (17820*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi in Sympy [A]  time = 42.8693, size = 136, normalized size = 0.89 \[ \frac{17820 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{738625 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{161051} - \frac{7554245}{5021863 \sqrt{- 2 x + 1}} - \frac{667615}{195657 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{19659}{1694 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{230}{121 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{5}{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17820*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 738625*sqrt(55)*atanh(sqr
t(55)*sqrt(-2*x + 1)/11)/161051 - 7554245/(5021863*sqrt(-2*x + 1)) - 667615/(195
657*(-2*x + 1)**(3/2)) + 19659/(1694*(-2*x + 1)**(3/2)*(3*x + 2)) + 230/(121*(-2
*x + 1)**(3/2)*(3*x + 2)*(5*x + 3)) - 5/(22*(-2*x + 1)**(3/2)*(3*x + 2)*(5*x + 3
)**2)

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Mathematica [A]  time = 0.174466, size = 109, normalized size = 0.72 \[ \frac{\sqrt{1-2 x} \left (6798820500 x^4+1580768100 x^3-4110847595 x^2-479695050 x+645558882\right )}{30131178 (3 x+2) \left (10 x^2+x-3\right )^2}+\frac{17820}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{738625 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(645558882 - 479695050*x - 4110847595*x^2 + 1580768100*x^3 + 6798
820500*x^4))/(30131178*(2 + 3*x)*(-3 + x + 10*x^2)^2) + (17820*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/14641

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Maple [A]  time = 0.026, size = 100, normalized size = 0.7 \[{\frac{32}{195657} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{5472}{5021863}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{162}{343}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{17820\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{156250}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{121}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1309}{250}\sqrt{1-2\,x}} \right ) }-{\frac{738625\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/195657/(1-2*x)^(3/2)+5472/5021863/(1-2*x)^(1/2)-162/343*(1-2*x)^(1/2)/(-4/3-2
*x)+17820/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+156250/14641*(-121/5
0*(1-2*x)^(3/2)+1309/250*(1-2*x)^(1/2))/(-6-10*x)^2-738625/161051*arctanh(1/11*5
5^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52025, size = 197, normalized size = 1.3 \[ \frac{738625}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8910}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1699705125 \,{\left (2 \, x - 1\right )}^{4} + 7589204550 \,{\left (2 \, x - 1\right )}^{3} + 8458535305 \,{\left (2 \, x - 1\right )}^{2} - 22225280 \, x + 13199648}{15065589 \,{\left (75 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 505 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 1133 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 847 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

738625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 8910/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 1/15065589*(1699705125*(2*x - 1)^4 + 7589204550*(2*x - 1)^3 +
8458535305*(2*x - 1)^2 - 22225280*x + 13199648)/(75*(-2*x + 1)^(9/2) - 505*(-2*x
+ 1)^(7/2) + 1133*(-2*x + 1)^(5/2) - 847*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.230678, size = 267, normalized size = 1.76 \[ \frac{\sqrt{11} \sqrt{7}{\left (760045125 \, \sqrt{7} \sqrt{5}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 782707860 \, \sqrt{11} \sqrt{3}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (6798820500 \, x^{4} + 1580768100 \, x^{3} - 4110847595 \, x^{2} - 479695050 \, x + 645558882\right )}\right )}}{2320100706 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2320100706*sqrt(11)*sqrt(7)*(760045125*sqrt(7)*sqrt(5)*(150*x^4 + 205*x^3 + 34
*x^2 - 51*x - 18)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x
+ 1))/(5*x + 3)) + 782707860*sqrt(11)*sqrt(3)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x
- 18)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x +
2)) - sqrt(11)*sqrt(7)*(6798820500*x^4 + 1580768100*x^3 - 4110847595*x^2 - 47969
5050*x + 645558882))/((150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*sqrt(-2*x + 1))

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221015, size = 194, normalized size = 1.28 \[ \frac{738625}{322102} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{8910}{2401} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{64 \,{\left (513 \, x - 295\right )}}{15065589 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{243 \, \sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} - \frac{625 \,{\left (55 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 119 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

738625/322102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5
*sqrt(-2*x + 1))) - 8910/2401*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/15065589*(513*x - 295)/((2*x - 1)*sqrt(-2*
x + 1)) + 243/343*sqrt(-2*x + 1)/(3*x + 2) - 625/5324*(55*(-2*x + 1)^(3/2) - 119
*sqrt(-2*x + 1))/(5*x + 3)^2