3.1980 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=124 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}+\frac{7103 \sqrt{1-2 x}}{30 (5 x+3)}-\frac{1133 \sqrt{1-2 x}}{30 (5 x+3)^2}+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7209}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1133*Sqrt[1 - 2*x])/(30*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5
*x)^2) + (7103*Sqrt[1 - 2*x])/(30*(3 + 5*x)) + 1400*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]] - (7209*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

_______________________________________________________________________________________

Rubi [A]  time = 0.256805, antiderivative size = 124, 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 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}+\frac{7103 \sqrt{1-2 x}}{30 (5 x+3)}-\frac{1133 \sqrt{1-2 x}}{30 (5 x+3)^2}+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{7209}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1133*Sqrt[1 - 2*x])/(30*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5
*x)^2) + (7103*Sqrt[1 - 2*x])/(30*(3 + 5*x)) + 1400*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]] - (7209*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.1644, size = 107, normalized size = 0.86 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{7103 \sqrt{- 2 x + 1}}{30 \left (5 x + 3\right )} - \frac{1133 \sqrt{- 2 x + 1}}{30 \left (5 x + 3\right )^{2}} + \frac{1400 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} - \frac{7209 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(3*(3*x + 2)*(5*x + 3)**2) + 7103*sqrt(-2*x + 1)/(30*(5*x +
3)) - 1133*sqrt(-2*x + 1)/(30*(5*x + 3)**2) + 1400*sqrt(21)*atanh(sqrt(21)*sqrt(
-2*x + 1)/7)/3 - 7209*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/25

_______________________________________________________________________________________

Mathematica [A]  time = 0.218613, size = 93, normalized size = 0.75 \[ \frac{1}{50} \left (\frac{5 \sqrt{1-2 x} \left (35515 x^2+43806 x+13474\right )}{(3 x+2) (5 x+3)^2}-14418 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )+1400 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

1400*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + ((5*Sqrt[1 - 2*x]*(13474 + 438
06*x + 35515*x^2))/((2 + 3*x)*(3 + 5*x)^2) - 14418*Sqrt[55]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]])/50

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 82, normalized size = 0.7 \[ -{\frac{98}{3}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{1400\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{141\, \left ( 1-2\,x \right ) ^{3/2}}{50}}+{\frac{1529\,\sqrt{1-2\,x}}{250}} \right ) }-{\frac{7209\,\sqrt{55}}{25}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-98/3*(1-2*x)^(1/2)/(-4/3-2*x)+1400/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/
2)+550*(-141/50*(1-2*x)^(3/2)+1529/250*(1-2*x)^(1/2))/(-6-10*x)^2-7209/25*arctan
h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.60069, size = 173, normalized size = 1.4 \[ \frac{7209}{50} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{700}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{35515 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 158642 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177023 \, \sqrt{-2 \, x + 1}}{5 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7209/50*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 700/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/5*(35515*(-2*x + 1)^(5/2) - 158642*(-2*x + 1)^(3/2) + 177023*sqrt(-2*x
+ 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)

_______________________________________________________________________________________

Fricas [A]  time = 0.243497, size = 213, normalized size = 1.72 \[ \frac{\sqrt{5} \sqrt{3}{\left (7209 \, \sqrt{11} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 7000 \, \sqrt{7} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{5} \sqrt{3}{\left (35515 \, x^{2} + 43806 \, x + 13474\right )} \sqrt{-2 \, x + 1}\right )}}{150 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/150*sqrt(5)*sqrt(3)*(7209*sqrt(11)*sqrt(3)*(75*x^3 + 140*x^2 + 87*x + 18)*log(
(sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 7000*sqrt(7)*sqrt(5
)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x +
1))/(3*x + 2)) + sqrt(5)*sqrt(3)*(35515*x^2 + 43806*x + 13474)*sqrt(-2*x + 1))/(
75*x^3 + 140*x^2 + 87*x + 18)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215712, size = 166, normalized size = 1.34 \[ \frac{7209}{50} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{700}{3} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{49 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{11 \,{\left (705 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1529 \, \sqrt{-2 \, x + 1}\right )}}{20 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7209/50*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 700/3*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 49*sqrt(-2*x + 1)/(3*x + 2) - 11/20*(705*(-2*x + 1)^(3/
2) - 1529*sqrt(-2*x + 1))/(5*x + 3)^2