3.2162 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=145 \[ -\frac{446660}{290521 \sqrt{1-2 x}}+\frac{582}{49 (1-2 x)^{3/2} (3 x+2)}-\frac{39520}{11319 (1-2 x)^{3/2}}+\frac{57}{49 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{7 (1-2 x)^{3/2} (3 x+2)^3}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-39520/(11319*(1 - 2*x)^(3/2)) - 446660/(290521*Sqrt[1 - 2*x]) + 1/(7*(1 - 2*x)^
(3/2)*(2 + 3*x)^3) + 57/(49*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 582/(49*(1 - 2*x)^(3/
2)*(2 + 3*x)) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2401 - (6250
*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.422088, antiderivative size = 145, 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 \[ -\frac{446660}{290521 \sqrt{1-2 x}}+\frac{582}{49 (1-2 x)^{3/2} (3 x+2)}-\frac{39520}{11319 (1-2 x)^{3/2}}+\frac{57}{49 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{7 (1-2 x)^{3/2} (3 x+2)^3}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-39520/(11319*(1 - 2*x)^(3/2)) - 446660/(290521*Sqrt[1 - 2*x]) + 1/(7*(1 - 2*x)^
(3/2)*(2 + 3*x)^3) + 57/(49*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 582/(49*(1 - 2*x)^(3/
2)*(2 + 3*x)) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2401 - (6250
*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 41.5166, size = 128, normalized size = 0.88 \[ \frac{127710 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{16807} - \frac{6250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} - \frac{446660}{290521 \sqrt{- 2 x + 1}} - \frac{39520}{11319 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{582}{49 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{57}{49 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} + \frac{1}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

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),x)

[Out]

127710*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/16807 - 6250*sqrt(55)*atanh(sqr
t(55)*sqrt(-2*x + 1)/11)/1331 - 446660/(290521*sqrt(-2*x + 1)) - 39520/(11319*(-
2*x + 1)**(3/2)) + 582/(49*(-2*x + 1)**(3/2)*(3*x + 2)) + 57/(49*(-2*x + 1)**(3/
2)*(3*x + 2)**2) + 1/(7*(-2*x + 1)**(3/2)*(3*x + 2)**3)

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Mathematica [A]  time = 0.219707, size = 99, normalized size = 0.68 \[ \frac{72358920 x^4+26376300 x^3-47036214 x^2-9083055 x+8496203}{871563 (1-2 x)^{3/2} (3 x+2)^3}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(8496203 - 9083055*x - 47036214*x^2 + 26376300*x^3 + 72358920*x^4)/(871563*(1 -
2*x)^(3/2)*(2 + 3*x)^3) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24
01 - (6250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.024, size = 93, normalized size = 0.6 \[{\frac{32}{79233} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{5344}{2033647}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{1458}{16807\, \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1438}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{61250}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{72520}{27}\sqrt{1-2\,x}} \right ) }+{\frac{127710\,\sqrt{21}}{16807}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{6250\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/79233/(1-2*x)^(3/2)+5344/2033647/(1-2*x)^(1/2)-1458/16807*(1438/3*(1-2*x)^(5/
2)-61250/27*(1-2*x)^(3/2)+72520/27*(1-2*x)^(1/2))/(-4-6*x)^3+127710/16807*arctan
h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(
1/2))*55^(1/2)

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Maxima [A]  time = 1.50279, size = 197, normalized size = 1.36 \[ \frac{3125}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{63855}{16807} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (9044865 \,{\left (2 \, x - 1\right )}^{4} + 42773535 \,{\left (2 \, x - 1\right )}^{3} + 50533308 \,{\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 63855/16807*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 4/871563*(9044865*(2*x - 1)^4 + 42773535*(2*x - 1)^3 + 50533308*
(2*x - 1)^2 - 315168*x + 187768)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 4
41*(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.227926, size = 267, normalized size = 1.84 \[ \frac{\sqrt{11} \sqrt{7}{\left (22509375 \, \sqrt{7} \sqrt{5}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 23179365 \, \sqrt{11} \sqrt{3}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (72358920 \, x^{4} + 26376300 \, x^{3} - 47036214 \, x^{2} - 9083055 \, x + 8496203\right )}\right )}}{67110351 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/67110351*sqrt(11)*sqrt(7)*(22509375*sqrt(7)*sqrt(5)*(54*x^4 + 81*x^3 + 18*x^2
- 20*x - 8)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/
(5*x + 3)) + 23179365*sqrt(11)*sqrt(3)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqr
t(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt
(11)*sqrt(7)*(72358920*x^4 + 26376300*x^3 - 47036214*x^2 - 9083055*x + 8496203))
/((54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*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)**4/(3+5*x),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221383, size = 181, normalized size = 1.25 \[ \frac{3125}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{63855}{16807} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{4 \,{\left (9044865 \,{\left (2 \, x - 1\right )}^{4} + 42773535 \,{\left (2 \, x - 1\right )}^{3} + 50533308 \,{\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3125/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 63855/16807*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/
(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/871563*(9044865*(2*x - 1)^4 + 42773535*(2*x -
1)^3 + 50533308*(2*x - 1)^2 - 315168*x + 187768)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-
2*x + 1))^3