3.2117 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=166 \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

-15987390/(456533*Sqrt[1 - 2*x]) - 35825/(1078*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(1
4*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2) + 435/(98*Sqrt[1 - 2*x]*(2 + 3*x)*(3 +
5*x)^2) + 1176400/(5929*Sqrt[1 - 2*x]*(3 + 5*x)) + (414315*Sqrt[3/7]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/343 - (1561125*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/1331

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Rubi [A]  time = 0.428784, antiderivative size = 166, 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 \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

-15987390/(456533*Sqrt[1 - 2*x]) - 35825/(1078*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(1
4*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2) + 435/(98*Sqrt[1 - 2*x]*(2 + 3*x)*(3 +
5*x)^2) + 1176400/(5929*Sqrt[1 - 2*x]*(3 + 5*x)) + (414315*Sqrt[3/7]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/343 - (1561125*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/1331

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Rubi in Sympy [A]  time = 42.7566, size = 146, normalized size = 0.88 \[ \frac{414315 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{1561125 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} - \frac{15987390}{456533 \sqrt{- 2 x + 1}} + \frac{705840}{5929 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{20067}{1694 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{635}{242 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{5}{22 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

414315*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 1561125*sqrt(55)*atanh(s
qrt(55)*sqrt(-2*x + 1)/11)/14641 - 15987390/(456533*sqrt(-2*x + 1)) + 705840/(59
29*sqrt(-2*x + 1)*(3*x + 2)) + 20067/(1694*sqrt(-2*x + 1)*(3*x + 2)**2) + 635/(2
42*sqrt(-2*x + 1)*(3*x + 2)**2*(5*x + 3)) - 5/(22*sqrt(-2*x + 1)*(3*x + 2)**2*(5
*x + 3)**2)

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Mathematica [A]  time = 0.199375, size = 106, normalized size = 0.64 \[ \frac{-7194325500 x^4-10073172600 x^3-1810042755 x^2+2503057145 x+909821467}{913066 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

(909821467 + 2503057145*x - 1810042755*x^2 - 10073172600*x^3 - 7194325500*x^4)/(
913066*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2) + (414315*Sqrt[3/7]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/343 - (1561125*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/1331

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Maple [A]  time = 0.024, size = 103, normalized size = 0.6 \[{\frac{64}{456533}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{8748}{343\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{217}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{414315\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{312500}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{191}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2079}{500}\sqrt{1-2\,x}} \right ) }-{\frac{1561125\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64/456533/(1-2*x)^(1/2)-8748/343*(217/36*(1-2*x)^(3/2)-511/36*(1-2*x)^(1/2))/(-4
-6*x)^2+414315/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+312500/1331*(-1
91/100*(1-2*x)^(3/2)+2079/500*(1-2*x)^(1/2))/(-6-10*x)^2-1561125/14641*arctanh(1
/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.4962, size = 209, normalized size = 1.26 \[ \frac{1561125}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{414315}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1798581375 \,{\left (2 \, x - 1\right )}^{4} + 12230911800 \,{\left (2 \, x - 1\right )}^{3} + 27711289905 \,{\left (2 \, x - 1\right )}^{2} + 41836111240 \, x - 20918245348\right )}}{456533 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2040 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 6934 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10472 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 5929 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1561125/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 414315/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) - 2/456533*(1798581375*(2*x - 1)^4 + 12230911800*(2*x - 1)^3 +
27711289905*(2*x - 1)^2 + 41836111240*x - 20918245348)/(225*(-2*x + 1)^(9/2) -
2040*(-2*x + 1)^(7/2) + 6934*(-2*x + 1)^(5/2) - 10472*(-2*x + 1)^(3/2) + 5929*sq
rt(-2*x + 1))

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Fricas [A]  time = 0.233762, size = 267, normalized size = 1.61 \[ \frac{\sqrt{11} \sqrt{7}{\left (535465875 \, \sqrt{7} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 551453265 \, \sqrt{11} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (7194325500 \, x^{4} + 10073172600 \, x^{3} + 1810042755 \, x^{2} - 2503057145 \, x - 909821467\right )}\right )}}{70306082 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/70306082*sqrt(11)*sqrt(7)*(535465875*sqrt(7)*sqrt(5)*(225*x^4 + 570*x^3 + 541*
x^2 + 228*x + 36)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x
+ 1))/(5*x + 3)) + 551453265*sqrt(11)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228
*x + 36)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x
+ 2)) - sqrt(11)*sqrt(7)*(7194325500*x^4 + 10073172600*x^3 + 1810042755*x^2 - 25
03057145*x - 909821467))/((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*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)**3/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.241922, size = 212, normalized size = 1.28 \[ \frac{1561125}{29282} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{414315}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{64}{456533 \, \sqrt{-2 \, x + 1}} + \frac{2 \,{\left (256941225 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1747282440 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 3958787399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2988341532 \, \sqrt{-2 \, x + 1}\right )}}{65219 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1561125/29282*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5
*sqrt(-2*x + 1))) - 414315/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x +
1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/456533/sqrt(-2*x + 1) + 2/65219*(2569412
25*(2*x - 1)^3*sqrt(-2*x + 1) + 1747282440*(2*x - 1)^2*sqrt(-2*x + 1) - 39587873
99*(-2*x + 1)^(3/2) + 2988341532*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2