3.2116 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=139 \[ -\frac{224967}{65219 \sqrt{1-2 x}}+\frac{33115}{1694 \sqrt{1-2 x} (5 x+3)}-\frac{505}{154 \sqrt{1-2 x} (5 x+3)^2}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{153825 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

-224967/(65219*Sqrt[1 - 2*x]) - 505/(154*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(7*Sqrt[
1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + 33115/(1694*Sqrt[1 - 2*x]*(3 + 5*x)) + (5832*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (153825*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.35156, antiderivative size = 139, 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 \[ -\frac{224967}{65219 \sqrt{1-2 x}}+\frac{33115}{1694 \sqrt{1-2 x} (5 x+3)}-\frac{505}{154 \sqrt{1-2 x} (5 x+3)^2}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{153825 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

-224967/(65219*Sqrt[1 - 2*x]) - 505/(154*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(7*Sqrt[
1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + 33115/(1694*Sqrt[1 - 2*x]*(3 + 5*x)) + (5832*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (153825*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/1331

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Rubi in Sympy [A]  time = 35.2166, size = 124, normalized size = 0.89 \[ \frac{5832 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{153825 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} - \frac{224967}{65219 \sqrt{- 2 x + 1}} + \frac{19869}{1694 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{235}{121 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{5}{22 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5832*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 153825*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)/14641 - 224967/(65219*sqrt(-2*x + 1)) + 19869/(1694*sqrt(
-2*x + 1)*(3*x + 2)) + 235/(121*sqrt(-2*x + 1)*(3*x + 2)*(5*x + 3)) - 5/(22*sqrt
(-2*x + 1)*(3*x + 2)*(5*x + 3)**2)

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Mathematica [A]  time = 0.21929, size = 103, normalized size = 0.74 \[ \frac{\frac{11 \sqrt{1-2 x} \left (33745050 x^3+24742935 x^2-8019782 x-6400750\right )}{(5 x+3)^2 \left (6 x^2+x-2\right )}-15074850 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1434818}+\frac{5832}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(5832*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((11*Sqrt[1 - 2*x]*(-6400
750 - 8019782*x + 24742935*x^2 + 33745050*x^3))/((3 + 5*x)^2*(-2 + x + 6*x^2)) -
15074850*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1434818

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Maple [A]  time = 0.024, size = 91, normalized size = 0.7 \[{\frac{32}{65219}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{54}{49}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{5832\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{31250}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{5}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1353}{250}\sqrt{1-2\,x}} \right ) }-{\frac{153825\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/65219/(1-2*x)^(1/2)-54/49*(1-2*x)^(1/2)/(-4/3-2*x)+5832/343*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)+31250/1331*(-5/2*(1-2*x)^(3/2)+1353/250*(1-2*x)^(1/2
))/(-6-10*x)^2-153825/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49121, size = 185, normalized size = 1.33 \[ \frac{153825}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2916}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{16872525 \,{\left (2 \, x - 1\right )}^{3} + 75360510 \,{\left (2 \, x - 1\right )}^{2} + 168127762 \, x - 84090985}{65219 \,{\left (75 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 505 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1133 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 847 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153825/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 2916/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 1/65219*(16872525*(2*x - 1)^3 + 75360510*(2*x - 1)^2 + 168127762
*x - 84090985)/(75*(-2*x + 1)^(7/2) - 505*(-2*x + 1)^(5/2) + 1133*(-2*x + 1)^(3/
2) - 847*sqrt(-2*x + 1))

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Fricas [A]  time = 0.249305, size = 240, normalized size = 1.73 \[ \frac{\sqrt{11} \sqrt{7}{\left (7537425 \, \sqrt{7} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 7762392 \, \sqrt{11} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (33745050 \, x^{3} + 24742935 \, x^{2} - 8019782 \, x - 6400750\right )}\right )}}{10043726 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10043726*sqrt(11)*sqrt(7)*(7537425*sqrt(7)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x +
18)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3
)) + 7762392*sqrt(11)*sqrt(3)*(75*x^3 + 140*x^2 + 87*x + 18)*sqrt(-2*x + 1)*log(
(sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(33
745050*x^3 + 24742935*x^2 - 8019782*x - 6400750))/((75*x^3 + 140*x^2 + 87*x + 18
)*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)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.2271, size = 182, normalized size = 1.31 \[ \frac{153825}{29282} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2916}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (215526 \, x - 107875\right )}}{65219 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} - \frac{125 \,{\left (625 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1353 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153825/29282*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*
sqrt(-2*x + 1))) - 2916/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/
(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/65219*(215526*x - 107875)/(3*(-2*x + 1)^(3/2)
- 7*sqrt(-2*x + 1)) - 125/5324*(625*(-2*x + 1)^(3/2) - 1353*sqrt(-2*x + 1))/(5*
x + 3)^2