3.2170 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=105 \[ \frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{218}{2541 (1-2 x)^{3/2}}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1400 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

218/(2541*(1 - 2*x)^(3/2)) + 3274/(65219*Sqrt[1 - 2*x]) - 5/(11*(1 - 2*x)^(3/2)*
(3 + 5*x)) - (54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*Sqrt[5/1
1]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.278468, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{218}{2541 (1-2 x)^{3/2}}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1400 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

218/(2541*(1 - 2*x)^(3/2)) + 3274/(65219*Sqrt[1 - 2*x]) - 5/(11*(1 - 2*x)^(3/2)*
(3 + 5*x)) - (54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*Sqrt[5/1
1]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 27.9169, size = 90, normalized size = 0.86 \[ - \frac{54 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} + \frac{1400 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{3274}{65219 \sqrt{- 2 x + 1}} + \frac{218}{2541 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{5}{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 + 1400*sqrt(55)*atanh(sqrt(55)
*sqrt(-2*x + 1)/11)/14641 + 3274/(65219*sqrt(-2*x + 1)) + 218/(2541*(-2*x + 1)**
(3/2)) - 5/(11*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.244139, size = 88, normalized size = 0.84 \[ \frac{205800 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{11 \left (98220 x^2-74108 x+9111\right )}{(1-2 x)^{3/2} (5 x+3)}}{2152227}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((-11*(9111 - 74108*x + 98
220*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) + 205800*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[
1 - 2*x]])/2152227

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Maple [A]  time = 0.021, size = 72, normalized size = 0.7 \[{\frac{8}{2541} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{824}{65219}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{54\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{50}{1331}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{1400\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/2541/(1-2*x)^(3/2)+824/65219/(1-2*x)^(1/2)-54/343*arctanh(1/7*21^(1/2)*(1-2*x)
^(1/2))*21^(1/2)+50/1331*(1-2*x)^(1/2)/(-6/5-2*x)+1400/14641*arctanh(1/11*55^(1/
2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50781, size = 149, normalized size = 1.42 \[ -\frac{700}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{27}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (24555 \,{\left (2 \, x - 1\right )}^{2} + 24112 \, x - 15444\right )}}{195657 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-700/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 27/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 2/195657*(24555*(2*x - 1)^2 + 24112*x - 15444)/(5*(-2*x + 1)^(5/2) -
11*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.240011, size = 204, normalized size = 1.94 \[ \frac{\sqrt{11} \sqrt{7}{\left (102900 \, \sqrt{7} \sqrt{5}{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 107811 \, \sqrt{11} \sqrt{3}{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (98220 \, x^{2} - 74108 \, x + 9111\right )}\right )}}{15065589 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15065589*sqrt(11)*sqrt(7)*(102900*sqrt(7)*sqrt(5)*(10*x^2 + x - 3)*sqrt(-2*x +
1)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 107811*sqr
t(11)*sqrt(3)*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)
*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(98220*x^2 - 74108*x + 9111))/((1
0*x^2 + x - 3)*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+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220283, size = 157, normalized size = 1.5 \[ -\frac{700}{14641} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{27}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (309 \, x - 193\right )}}{195657 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{125 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-700/14641*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 27/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 16/195657*(309*x - 193)/((2*x - 1)*sqrt(-2*x + 1))
- 125/1331*sqrt(-2*x + 1)/(5*x + 3)