3.2049 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=126 \[ \frac{33465 \sqrt{1-2 x}}{1694 (5 x+3)}-\frac{505 \sqrt{1-2 x}}{154 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{1908}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{32025}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-505*Sqrt[1 - 2*x])/(154*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x
)^2) + (33465*Sqrt[1 - 2*x])/(1694*(3 + 5*x)) + (1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/7 - (32025*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.26613, antiderivative size = 126, 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 \[ \frac{33465 \sqrt{1-2 x}}{1694 (5 x+3)}-\frac{505 \sqrt{1-2 x}}{154 (5 x+3)^2}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{1908}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{32025}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-505*Sqrt[1 - 2*x])/(154*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x
)^2) + (33465*Sqrt[1 - 2*x])/(1694*(3 + 5*x)) + (1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/7 - (32025*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 27.6392, size = 110, normalized size = 0.87 \[ \frac{20079 \sqrt{- 2 x + 1}}{1694 \left (3 x + 2\right )} + \frac{240 \sqrt{- 2 x + 1}}{121 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{5 \sqrt{- 2 x + 1}}{22 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{1908 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{32025 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20079*sqrt(-2*x + 1)/(1694*(3*x + 2)) + 240*sqrt(-2*x + 1)/(121*(3*x + 2)*(5*x +
3)) - 5*sqrt(-2*x + 1)/(22*(3*x + 2)*(5*x + 3)**2) + 1908*sqrt(21)*atanh(sqrt(2
1)*sqrt(-2*x + 1)/7)/49 - 32025*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1331

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Mathematica [A]  time = 0.237878, size = 95, normalized size = 0.75 \[ \frac{\frac{11 \sqrt{1-2 x} \left (501975 x^2+619170 x+190406\right )}{(3 x+2) (5 x+3)^2}-448350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18634}+\frac{1908}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((11*Sqrt[1 - 2*x]*(190406
+ 619170*x + 501975*x^2))/((2 + 3*x)*(3 + 5*x)^2) - 448350*Sqrt[55]*ArcTanh[Sqr
t[5/11]*Sqrt[1 - 2*x]])/18634

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Maple [A]  time = 0.02, size = 82, normalized size = 0.7 \[ -{\frac{18}{7}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{1908\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+1250\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{129\, \left ( 1-2\,x \right ) ^{3/2}}{1210}}+{\frac{127\,\sqrt{1-2\,x}}{550}} \right ) }-{\frac{32025\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18/7*(1-2*x)^(1/2)/(-4/3-2*x)+1908/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1
/2)+1250*(-129/1210*(1-2*x)^(3/2)+127/550*(1-2*x)^(1/2))/(-6-10*x)^2-32025/1331*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50091, size = 173, normalized size = 1.37 \[ \frac{32025}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{954}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{501975 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 2242290 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2501939 \, \sqrt{-2 \, x + 1}}{847 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32025/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 954/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 1/847*(501975*(-2*x + 1)^(5/2) - 2242290*(-2*x + 1)^(3/2) + 2501939*
sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)

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Fricas [A]  time = 0.220382, size = 213, normalized size = 1.69 \[ \frac{\sqrt{11} \sqrt{7}{\left (224175 \, \sqrt{7} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 230868 \, \sqrt{11} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (501975 \, x^{2} + 619170 \, x + 190406\right )} \sqrt{-2 \, x + 1}\right )}}{130438 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/130438*sqrt(11)*sqrt(7)*(224175*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)) + 230868*sqrt(1
1)*sqrt(3)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqr
t(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(501975*x^2 + 619170*x + 190406)*sqrt
(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.234148, size = 166, normalized size = 1.32 \[ \frac{32025}{2662} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{954}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{27 \, \sqrt{-2 \, x + 1}}{7 \,{\left (3 \, x + 2\right )}} - \frac{25 \,{\left (645 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1397 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32025/2662*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) - 954/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 27/7*sqrt(-2*x + 1)/(3*x + 2) - 25/484*(645*(-2*x +
1)^(3/2) - 1397*sqrt(-2*x + 1))/(5*x + 3)^2