3.2048 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{315 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{5 \sqrt{1-2 x}}{22 (5 x+3)^2}+18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2115}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-5*Sqrt[1 - 2*x])/(22*(3 + 5*x)^2) + (315*Sqrt[1 - 2*x])/(242*(3 + 5*x)) + 18*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2115*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.204904, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{315 \sqrt{1-2 x}}{242 (5 x+3)}-\frac{5 \sqrt{1-2 x}}{22 (5 x+3)^2}+18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2115}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x])/(22*(3 + 5*x)^2) + (315*Sqrt[1 - 2*x])/(242*(3 + 5*x)) + 18*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2115*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 20.5248, size = 83, normalized size = 0.86 \[ \frac{315 \sqrt{- 2 x + 1}}{242 \left (5 x + 3\right )} - \frac{5 \sqrt{- 2 x + 1}}{22 \left (5 x + 3\right )^{2}} + \frac{18 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} - \frac{2115 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

315*sqrt(-2*x + 1)/(242*(5*x + 3)) - 5*sqrt(-2*x + 1)/(22*(5*x + 3)**2) + 18*sqr
t(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7 - 2115*sqrt(55)*atanh(sqrt(55)*sqrt(-2*
x + 1)/11)/1331

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Mathematica [A]  time = 0.233871, size = 81, normalized size = 0.84 \[ 18 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{5 \left (\frac{11 \sqrt{1-2 x} (315 x+178)}{(5 x+3)^2}-846 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{2662} \]

Antiderivative was successfully verified.

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

[Out]

18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (5*((11*Sqrt[1 - 2*x]*(178 + 315
*x))/(3 + 5*x)^2 - 846*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/2662

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Maple [A]  time = 0.018, size = 66, normalized size = 0.7 \[{\frac{18\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+250\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{63\, \left ( 1-2\,x \right ) ^{3/2}}{1210}}+{\frac{61\,\sqrt{1-2\,x}}{550}} \right ) }-{\frac{2115\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+250*(-63/1210*(1-2*x)^(3/2)+61
/550*(1-2*x)^(1/2))/(-6-10*x)^2-2115/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*5
5^(1/2)

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Maxima [A]  time = 1.4894, size = 149, normalized size = 1.54 \[ \frac{2115}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5 \,{\left (315 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 671 \, \sqrt{-2 \, x + 1}\right )}}{121 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2115/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 9/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 5/121*(315*(-2*x + 1)^(3/2) - 671*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*
x + 11)

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Fricas [A]  time = 0.229124, size = 188, normalized size = 1.94 \[ \frac{\sqrt{11} \sqrt{7}{\left (2115 \, \sqrt{7} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2178 \, \sqrt{11} \sqrt{3}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 5 \, \sqrt{11} \sqrt{7}{\left (315 \, x + 178\right )} \sqrt{-2 \, x + 1}\right )}}{18634 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18634*sqrt(11)*sqrt(7)*(2115*sqrt(7)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)
*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 2178*sqrt(11)*sqrt(3)*(25*x
^2 + 30*x + 9)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + 5
*sqrt(11)*sqrt(7)*(315*x + 178)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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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/(2+3*x)/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.272268, size = 144, normalized size = 1.48 \[ \frac{2115}{2662} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9}{7} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5 \,{\left (315 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 671 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2115/2662*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 9/7*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) - 5/484*(315*(-2*x + 1)^(3/2) - 671*sqrt(-2*x + 1))/(5*x
+ 3)^2