3.2106 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=119 \[ \frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (5 x+3)}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

4644/(5929*Sqrt[1 - 2*x]) - 340/(77*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(7*Sqrt[1 - 2*x
]*(2 + 3*x)*(3 + 5*x)) - (1314*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 +
(3150*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.280965, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (5 x+3)}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4644/(5929*Sqrt[1 - 2*x]) - 340/(77*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(7*Sqrt[1 - 2*x
]*(2 + 3*x)*(3 + 5*x)) - (1314*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 +
(3150*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 28.2827, size = 100, normalized size = 0.84 \[ - \frac{1314 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} + \frac{3150 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{4644}{5929 \sqrt{- 2 x + 1}} - \frac{204}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{5}{11 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1314*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 + 3150*sqrt(55)*atanh(sqrt(5
5)*sqrt(-2*x + 1)/11)/1331 + 4644/(5929*sqrt(-2*x + 1)) - 204/(77*sqrt(-2*x + 1)
*(3*x + 2)) - 5/(11*sqrt(-2*x + 1)*(3*x + 2)*(5*x + 3))

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Mathematica [A]  time = 0.312086, size = 96, normalized size = 0.81 \[ \frac{69660 x^2+9696 x-21955}{5929 \sqrt{1-2 x} (3 x+2) (5 x+3)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-21955 + 9696*x + 69660*x^2)/(5929*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) - (1314*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (3150*Sqrt[5/11]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.023, size = 79, normalized size = 0.7 \[{\frac{16}{5929}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{18}{49}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{1314\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{50}{121}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{3150\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/5929/(1-2*x)^(1/2)+18/49*(1-2*x)^(1/2)/(-4/3-2*x)-1314/343*arctanh(1/7*21^(1/
2)*(1-2*x)^(1/2))*21^(1/2)+50/121*(1-2*x)^(1/2)/(-6/5-2*x)+3150/1331*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49537, size = 161, normalized size = 1.35 \[ -\frac{1575}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{657}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (17415 \,{\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 77 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1575/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 657/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 + 79356*x - 39370)/(15*(-2*x + 1)^(5/2) -
68*(-2*x + 1)^(3/2) + 77*sqrt(-2*x + 1))

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Fricas [A]  time = 0.242777, size = 212, normalized size = 1.78 \[ \frac{\sqrt{11} \sqrt{7}{\left (77175 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 79497 \, \sqrt{11} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (69660 \, x^{2} + 9696 \, x - 21955\right )}\right )}}{456533 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/456533*sqrt(11)*sqrt(7)*(77175*sqrt(7)*sqrt(5)*(15*x^2 + 19*x + 6)*sqrt(-2*x +
1)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 79497*sqrt
(11)*sqrt(3)*(15*x^2 + 19*x + 6)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(
3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(69660*x^2 + 9696*x - 21955))/(
(15*x^2 + 19*x + 6)*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)**(3/2)/(2+3*x)**2/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221376, size = 178, normalized size = 1.5 \[ -\frac{1575}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{657}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (17415 \,{\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 77 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1575/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 657/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 + 79356*x - 39370)/(15*(
2*x - 1)^2*sqrt(-2*x + 1) - 68*(-2*x + 1)^(3/2) + 77*sqrt(-2*x + 1))