3.2180 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=125 \[ -\frac{65167}{717409 \sqrt{1-2 x}}+\frac{295}{242 (1-2 x)^{3/2} (5 x+3)}-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{5}{22 (1-2 x)^{3/2} (5 x+3)^2}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{47075 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

-5969/(27951*(1 - 2*x)^(3/2)) - 65167/(717409*Sqrt[1 - 2*x]) - 5/(22*(1 - 2*x)^(
3/2)*(3 + 5*x)^2) + 295/(242*(1 - 2*x)^(3/2)*(3 + 5*x)) + (162*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (47075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/14641

_______________________________________________________________________________________

Rubi [A]  time = 0.345555, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{65167}{717409 \sqrt{1-2 x}}+\frac{295}{242 (1-2 x)^{3/2} (5 x+3)}-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{5}{22 (1-2 x)^{3/2} (5 x+3)^2}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{47075 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

-5969/(27951*(1 - 2*x)^(3/2)) - 65167/(717409*Sqrt[1 - 2*x]) - 5/(22*(1 - 2*x)^(
3/2)*(3 + 5*x)^2) + 295/(242*(1 - 2*x)^(3/2)*(3 + 5*x)) + (162*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (47075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/14641

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.5479, size = 109, normalized size = 0.87 \[ \frac{162 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{47075 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{161051} - \frac{65167}{717409 \sqrt{- 2 x + 1}} - \frac{5969}{27951 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{295}{242 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} - \frac{5}{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

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)**3,x)

[Out]

162*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 47075*sqrt(55)*atanh(sqrt(55
)*sqrt(-2*x + 1)/11)/161051 - 65167/(717409*sqrt(-2*x + 1)) - 5969/(27951*(-2*x
+ 1)**(3/2)) + 295/(242*(-2*x + 1)**(3/2)*(5*x + 3)) - 5/(22*(-2*x + 1)**(3/2)*(
5*x + 3)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.203798, size = 96, normalized size = 0.77 \[ \frac{\frac{11 \sqrt{1-2 x} \left (19550100 x^3-9295580 x^2-6032979 x+2971158\right )}{\left (10 x^2+x-3\right )^2}-13840050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{47348994}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((11*Sqrt[1 - 2*x]*(297115
8 - 6032979*x - 9295580*x^2 + 19550100*x^3))/(-3 + x + 10*x^2)^2 - 13840050*Sqrt
[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/47348994

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 84, normalized size = 0.7 \[{\frac{16}{27951} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2208}{717409}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{162\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{31250}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{11}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{583}{250}\sqrt{1-2\,x}} \right ) }-{\frac{47075\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/27951/(1-2*x)^(3/2)+2208/717409/(1-2*x)^(1/2)+162/343*arctanh(1/7*21^(1/2)*(1
-2*x)^(1/2))*21^(1/2)+31250/14641*(-11/10*(1-2*x)^(3/2)+583/250*(1-2*x)^(1/2))/(
-6-10*x)^2-47075/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51123, size = 173, normalized size = 1.38 \[ \frac{47075}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{81}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4887525 \,{\left (2 \, x - 1\right )}^{3} + 10014785 \,{\left (2 \, x - 1\right )}^{2} - 1331968 \, x + 815056}{2152227 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

47075/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 81/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/2152227*(4887525*(2*x - 1)^3 + 10014785*(2*x - 1)^2 - 1331968*x
+ 815056)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.234863, size = 240, normalized size = 1.92 \[ \frac{\sqrt{11} \sqrt{7}{\left (6920025 \, \sqrt{7} \sqrt{5}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 7115526 \, \sqrt{11} \sqrt{3}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (19550100 \, x^{3} - 9295580 \, x^{2} - 6032979 \, x + 2971158\right )}\right )}}{331442958 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/331442958*sqrt(11)*sqrt(7)*(6920025*sqrt(7)*sqrt(5)*(50*x^3 + 35*x^2 - 12*x -
9)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)
) + 7115526*sqrt(11)*sqrt(3)*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log((sq
rt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(19550
100*x^3 - 9295580*x^2 - 6032979*x + 2971158))/((50*x^3 + 35*x^2 - 12*x - 9)*sqrt
(-2*x + 1))

_______________________________________________________________________________________

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)**3,x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225096, size = 173, normalized size = 1.38 \[ \frac{47075}{322102} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{81}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (828 \, x - 491\right )}}{2152227 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{125 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 53 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

47075/322102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) - 81/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) + 16/2152227*(828*x - 491)/((2*x - 1)*sqrt(-2*x + 1
)) - 125/5324*(25*(-2*x + 1)^(3/2) - 53*sqrt(-2*x + 1))/(5*x + 3)^2