3.2172 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=159 \[ \frac{172105}{65219 \sqrt{1-2 x}}+\frac{24}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{745}{22 (1-2 x)^{3/2} (5 x+3)}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}+\frac{15185}{2541 (1-2 x)^{3/2}}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{117500 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

15185/(2541*(1 - 2*x)^(3/2)) + 172105/(65219*Sqrt[1 - 2*x]) - 745/(22*(1 - 2*x)^
(3/2)*(3 + 5*x)) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + 24/(7*(1 - 2*x
)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])
/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.423756, antiderivative size = 159, 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 \[ \frac{172105}{65219 \sqrt{1-2 x}}+\frac{24}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{745}{22 (1-2 x)^{3/2} (5 x+3)}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}+\frac{15185}{2541 (1-2 x)^{3/2}}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{117500 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

15185/(2541*(1 - 2*x)^(3/2)) + 172105/(65219*Sqrt[1 - 2*x]) - 745/(22*(1 - 2*x)^
(3/2)*(3 + 5*x)) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + 24/(7*(1 - 2*x
)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])
/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 41.9065, size = 133, normalized size = 0.84 \[ - \frac{4455 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} + \frac{117500 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{172105}{65219 \sqrt{- 2 x + 1}} + \frac{15185}{2541 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{447}{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} - \frac{309}{154 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} - \frac{5}{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4455*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 + 117500*sqrt(55)*atanh(sqrt
(55)*sqrt(-2*x + 1)/11)/14641 + 172105/(65219*sqrt(-2*x + 1)) + 15185/(2541*(-2*
x + 1)**(3/2)) - 447/(22*(-2*x + 1)**(3/2)*(3*x + 2)) - 309/(154*(-2*x + 1)**(3/
2)*(3*x + 2)**2) - 5/(11*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3))

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Mathematica [A]  time = 0.249426, size = 108, normalized size = 0.68 \[ \frac{34545000 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{11 \sqrt{1-2 x} \left (92936700 x^4+27977220 x^3-58371045 x^2-9008764 x+9784671\right )}{(5 x+3) \left (6 x^2+x-2\right )^2}}{4304454}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4455*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((-11*Sqrt[1 - 2*x]*(978
4671 - 9008764*x - 58371045*x^2 + 27977220*x^3 + 92936700*x^4))/((3 + 5*x)*(-2 +
x + 6*x^2)^2) + 34545000*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/4304454

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Maple [A]  time = 0.027, size = 100, normalized size = 0.6 \[{\frac{32}{124509} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{5408}{3195731}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4374}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{151}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{119}{6}\sqrt{1-2\,x}} \right ) }-{\frac{4455\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1250}{1331}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{117500\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/124509/(1-2*x)^(3/2)+5408/3195731/(1-2*x)^(1/2)+4374/2401*(151/18*(1-2*x)^(3/
2)-119/6*(1-2*x)^(1/2))/(-4-6*x)^2-4455/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*
21^(1/2)+1250/1331*(1-2*x)^(1/2)/(-6/5-2*x)+117500/14641*arctanh(1/11*55^(1/2)*(
1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50622, size = 197, normalized size = 1.24 \[ -\frac{58750}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4455}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{23234175 \,{\left (2 \, x - 1\right )}^{4} + 106925310 \,{\left (2 \, x - 1\right )}^{3} + 122999835 \,{\left (2 \, x - 1\right )}^{2} + 285824 \, x - 170016}{195657 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 309 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 707 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 539 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-58750/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 4455/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 1/195657*(23234175*(2*x - 1)^4 + 106925310*(2*x - 1)^3 + 1229998
35*(2*x - 1)^2 + 285824*x - 170016)/(45*(-2*x + 1)^(9/2) - 309*(-2*x + 1)^(7/2)
+ 707*(-2*x + 1)^(5/2) - 539*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.226455, size = 266, normalized size = 1.67 \[ \frac{\sqrt{11} \sqrt{7}{\left (17272500 \, \sqrt{7} \sqrt{5}{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 17788815 \, \sqrt{11} \sqrt{3}{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (92936700 \, x^{4} + 27977220 \, x^{3} - 58371045 \, x^{2} - 9008764 \, x + 9784671\right )}\right )}}{30131178 \,{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30131178*sqrt(11)*sqrt(7)*(17272500*sqrt(7)*sqrt(5)*(90*x^4 + 129*x^3 + 25*x^2
- 32*x - 12)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1)
)/(5*x + 3)) + 17788815*sqrt(11)*sqrt(3)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)
*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) +
sqrt(11)*sqrt(7)*(92936700*x^4 + 27977220*x^3 - 58371045*x^2 - 9008764*x + 97846
71))/((90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*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)**3/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230385, size = 194, normalized size = 1.22 \[ -\frac{58750}{14641} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4455}{686} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{64 \,{\left (507 \, x - 292\right )}}{9587193 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{3125 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} + \frac{243 \,{\left (151 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 357 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-58750/14641*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) + 4455/686*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/
(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/9587193*(507*x - 292)/((2*x - 1)*sqrt(-2*x +
1)) - 3125/1331*sqrt(-2*x + 1)/(5*x + 3) + 243/9604*(151*(-2*x + 1)^(3/2) - 357
*sqrt(-2*x + 1))/(3*x + 2)^2