3.1970 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}+\frac{343 \sqrt{1-2 x}}{9 (3 x+2) (5 x+3)}-\frac{6763 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-6763*Sqrt[1 - 2*x])/(18*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5
*x)) + (343*Sqrt[1 - 2*x])/(9*(2 + 3*x)*(3 + 5*x)) - (6665*Sqrt[7/3]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/3 + 2288*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.251381, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}+\frac{343 \sqrt{1-2 x}}{9 (3 x+2) (5 x+3)}-\frac{6763 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6763*Sqrt[1 - 2*x])/(18*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5
*x)) + (343*Sqrt[1 - 2*x])/(9*(2 + 3*x)*(3 + 5*x)) - (6665*Sqrt[7/3]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/3 + 2288*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 27.2124, size = 110, normalized size = 0.84 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{6763 \sqrt{- 2 x + 1}}{18 \left (5 x + 3\right )} + \frac{343 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{6665 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} + \frac{2288 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(6*(3*x + 2)**2*(5*x + 3)) - 6763*sqrt(-2*x + 1)/(18*(5*x +
3)) + 343*sqrt(-2*x + 1)/(9*(3*x + 2)*(5*x + 3)) - 6665*sqrt(21)*atanh(sqrt(21)*
sqrt(-2*x + 1)/7)/9 + 2288*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/5

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Mathematica [A]  time = 0.191953, size = 94, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (20289 x^2+26380 x+8553\right )}{6 (3 x+2)^2 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(8553 + 26380*x + 20289*x^2))/(6*(2 + 3*x)^2*(3 + 5*x)) - (6665*
Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + 2288*Sqrt[11/5]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 82, normalized size = 0.6 \[ 126\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{131\, \left ( 1-2\,x \right ) ^{3/2}}{18}}-{\frac{931\,\sqrt{1-2\,x}}{54}} \right ) }-{\frac{6665\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{5}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{2288\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

126*(131/18*(1-2*x)^(3/2)-931/54*(1-2*x)^(1/2))/(-4-6*x)^2-6665/9*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)+242/5*(1-2*x)^(1/2)/(-6/5-2*x)+2288/5*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52392, size = 173, normalized size = 1.32 \[ -\frac{1144}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6665}{18} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20289 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 93338 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 107261 \, \sqrt{-2 \, x + 1}}{3 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1144/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 6665/18*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/3*(20289*(-2*x + 1)^(5/2) - 93338*(-2*x + 1)^(3/2) + 107261*sqrt(-2*
x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

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Fricas [A]  time = 0.225562, size = 215, normalized size = 1.64 \[ \frac{\sqrt{5} \sqrt{3}{\left (6864 \, \sqrt{11} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 6665 \, \sqrt{7} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{5} \sqrt{3}{\left (20289 \, x^{2} + 26380 \, x + 8553\right )} \sqrt{-2 \, x + 1}\right )}}{90 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*sqrt(5)*sqrt(3)*(6864*sqrt(11)*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((s
qrt(5)*(5*x - 8) - 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 6665*sqrt(7)*sqrt(5)*
(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))
/(3*x + 2)) - sqrt(5)*sqrt(3)*(20289*x^2 + 26380*x + 8553)*sqrt(-2*x + 1))/(45*x
^3 + 87*x^2 + 56*x + 12)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216788, size = 166, normalized size = 1.27 \[ -\frac{1144}{5} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{6665}{18} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{121 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{7 \,{\left (393 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 931 \, \sqrt{-2 \, x + 1}\right )}}{12 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1144/5*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 6665/18*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) - 121*sqrt(-2*x + 1)/(5*x + 3) + 7/12*(393*(-2*x + 1)^(
3/2) - 931*sqrt(-2*x + 1))/(3*x + 2)^2