3.1830 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{48645 \sqrt{1-2 x}}{98 (5 x+3)}+\frac{7261 \sqrt{1-2 x}}{147 (3 x+2) (5 x+3)}+\frac{139 \sqrt{1-2 x}}{42 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac{335579}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+6650 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-48645*Sqrt[1 - 2*x])/(98*(3 + 5*x)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x))
+ (139*Sqrt[1 - 2*x])/(42*(2 + 3*x)^2*(3 + 5*x)) + (7261*Sqrt[1 - 2*x])/(147*(2
+ 3*x)*(3 + 5*x)) - (335579*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + 665
0*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.308064, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{48645 \sqrt{1-2 x}}{98 (5 x+3)}+\frac{7261 \sqrt{1-2 x}}{147 (3 x+2) (5 x+3)}+\frac{139 \sqrt{1-2 x}}{42 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac{335579}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+6650 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

(-48645*Sqrt[1 - 2*x])/(98*(3 + 5*x)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x))
+ (139*Sqrt[1 - 2*x])/(42*(2 + 3*x)^2*(3 + 5*x)) + (7261*Sqrt[1 - 2*x])/(147*(2
+ 3*x)*(3 + 5*x)) - (335579*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + 665
0*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.3641, size = 133, normalized size = 0.84 \[ - \frac{29187 \sqrt{- 2 x + 1}}{98 \left (3 x + 2\right )} - \frac{2095 \sqrt{- 2 x + 1}}{42 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{139 \sqrt{- 2 x + 1}}{42 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} + \frac{\sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{335579 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} + \frac{6650 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**2,x)

[Out]

-29187*sqrt(-2*x + 1)/(98*(3*x + 2)) - 2095*sqrt(-2*x + 1)/(42*(3*x + 2)*(5*x +
3)) + 139*sqrt(-2*x + 1)/(42*(3*x + 2)**2*(5*x + 3)) + sqrt(-2*x + 1)/(3*(3*x +
2)**3*(5*x + 3)) - 335579*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 + 6650*s
qrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.158113, size = 99, normalized size = 0.63 \[ -\frac{\sqrt{1-2 x} \left (1313415 x^3+2583264 x^2+1692159 x+369116\right )}{98 (3 x+2)^3 (5 x+3)}-\frac{335579}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+6650 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-(Sqrt[1 - 2*x]*(369116 + 1692159*x + 2583264*x^2 + 1313415*x^3))/(98*(2 + 3*x)^
3*(3 + 5*x)) - (335579*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + 6650*Sqr
t[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 91, normalized size = 0.6 \[ 324\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{7279\, \left ( 1-2\,x \right ) ^{5/2}}{588}}-{\frac{11023\, \left ( 1-2\,x \right ) ^{3/2}}{189}}+{\frac{7421\,\sqrt{1-2\,x}}{108}} \right ) }-{\frac{335579\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+50\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+{\frac{6650\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^2,x)

[Out]

324*(7279/588*(1-2*x)^(5/2)-11023/189*(1-2*x)^(3/2)+7421/108*(1-2*x)^(1/2))/(-4-
6*x)^3-335579/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+50*(1-2*x)^(1/2)/
(-6/5-2*x)+6650/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51996, size = 197, normalized size = 1.25 \[ -\frac{3325}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{335579}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1313415 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 9106773 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 21041937 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 16201507 \, \sqrt{-2 \, x + 1}}{49 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="maxima")

[Out]

-3325/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 335579/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/49*(1313415*(-2*x + 1)^(7/2) - 9106773*(-2*x + 1)^(5/2) + 210419
37*(-2*x + 1)^(3/2) - 16201507*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)
^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)

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Fricas [A]  time = 0.223085, size = 242, normalized size = 1.53 \[ \frac{\sqrt{11} \sqrt{7}{\left (325850 \, \sqrt{7} \sqrt{5}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 335579 \, \sqrt{11} \sqrt{3}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (1313415 \, x^{3} + 2583264 \, x^{2} + 1692159 \, x + 369116\right )} \sqrt{-2 \, x + 1}\right )}}{7546 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="fricas")

[Out]

1/7546*sqrt(11)*sqrt(7)*(325850*sqrt(7)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2 + 1
48*x + 24)*log((sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 335
579*sqrt(11)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(7)*(3*
x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(1313415*x^3 +
2583264*x^2 + 1692159*x + 369116)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 +
148*x + 24)

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Sympy [A]  time = 109.637, size = 658, normalized size = 4.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**2,x)

[Out]

-6060*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*
sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)
*sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) + 1632*Piecewise((sqrt(2
1)*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 +
1)/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/
7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x
+ 1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) - 336*Piecewise((sqrt(21)*(-5*l
og(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/32 -
5/(32*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)*
*2) - 1/(48*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(-2*x + 1)
/7 - 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(-2*x
+ 1)/7 - 1)**3))/7203, (x <= 1/2) & (x > -2/3))) - 5500*Piecewise((sqrt(55)*(-l
og(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1
/(4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/
605, (x <= 1/2) & (x > -3/5))) + 20100*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(
-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21
, -2*x + 1 < 7/3)) - 33500*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11
)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x +
1 < 11/5))

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GIAC/XCAS [A]  time = 0.227798, size = 188, normalized size = 1.19 \[ -\frac{3325}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{335579}{686} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{125 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{3 \,{\left (65511 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 308644 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 363629 \, \sqrt{-2 \, x + 1}\right )}}{392 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^2*(3*x + 2)^4),x, algorithm="giac")

[Out]

-3325/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 335579/686*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 125*sqrt(-2*x + 1)/(5*x + 3) - 3/392*(65511*(2*x
- 1)^2*sqrt(-2*x + 1) - 308644*(-2*x + 1)^(3/2) + 363629*sqrt(-2*x + 1))/(3*x +
2)^3