3.1837 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=121 \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-15*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^2) + (9
95*Sqrt[1 - 2*x])/(22*(3 + 5*x)) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.231447, antiderivative size = 121, 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 \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-15*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^2) + (9
95*Sqrt[1 - 2*x])/(22*(3 + 5*x)) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 27.5096, size = 104, normalized size = 0.86 \[ \frac{995 \sqrt{- 2 x + 1}}{22 \left (5 x + 3\right )} - \frac{15 \sqrt{- 2 x + 1}}{2 \left (5 x + 3\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{\left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{624 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} - \frac{6665 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

995*sqrt(-2*x + 1)/(22*(5*x + 3)) - 15*sqrt(-2*x + 1)/(2*(5*x + 3)**2) + sqrt(-2
*x + 1)/((3*x + 2)*(5*x + 3)**2) + 624*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)
/7 - 6665*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.178689, size = 94, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} \left (14925 x^2+18410 x+5662\right )}{22 (3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(5662 + 18410*x + 14925*x^2))/(22*(2 + 3*x)*(3 + 5*x)^2) + 624*Sq
rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]])/11

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Maple [A]  time = 0.02, size = 82, normalized size = 0.7 \[ -6\,{\frac{\sqrt{1-2\,x}}{-4/3-2\,x}}+{\frac{624\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+250\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{133\, \left ( 1-2\,x \right ) ^{3/2}}{110}}+{\frac{131\,\sqrt{1-2\,x}}{50}} \right ) }-{\frac{6665\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*(1-2*x)^(1/2)/(-4/3-2*x)+624/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2
50*(-133/110*(1-2*x)^(3/2)+131/50*(1-2*x)^(1/2))/(-6-10*x)^2-6665/121*arctanh(1/
11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49791, size = 173, normalized size = 1.43 \[ \frac{6665}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{312}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{14925 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 66670 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 74393 \, \sqrt{-2 \, x + 1}}{11 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6665/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 312/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1/11*(14925*(-2*x + 1)^(5/2) - 66670*(-2*x + 1)^(3/2) + 74393*sqrt(-2*x
+ 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)

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Fricas [A]  time = 0.220921, size = 213, normalized size = 1.76 \[ \frac{\sqrt{11} \sqrt{7}{\left (6665 \, \sqrt{7} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 6864 \, \sqrt{11} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (14925 \, x^{2} + 18410 \, x + 5662\right )} \sqrt{-2 \, x + 1}\right )}}{1694 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1694*sqrt(11)*sqrt(7)*(6665*sqrt(7)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x + 18)*log
((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 6864*sqrt(11)*sqr
t(3)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x
+ 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(14925*x^2 + 18410*x + 5662)*sqrt(-2*x + 1)
)/(75*x^3 + 140*x^2 + 87*x + 18)

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Sympy [A]  time = 72.4407, size = 468, normalized size = 3.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

252*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sq
rt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) + 1360*Piecewise((sqrt(55)
*(-log(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))) + 440*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqr
t(-2*x + 1)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/16 + 3/(16*(sqrt(
55)*sqrt(-2*x + 1)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)**2) + 3/(16
*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)**2))
/6655, (x <= 1/2) & (x > -3/5))) - 1854*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)/2
1, -2*x + 1 < 7/3)) + 3090*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 = 0.237369, size = 166, normalized size = 1.37 \[ \frac{6665}{242} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{312}{7} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{9 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{5 \,{\left (665 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1441 \, \sqrt{-2 \, x + 1}\right )}}{44 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6665/242*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 312/7*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) + 9*sqrt(-2*x + 1)/(3*x + 2) - 5/44*(665*(-2*x + 1)^(3/2
) - 1441*sqrt(-2*x + 1))/(5*x + 3)^2