3.1981 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=145 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac{2311 \sqrt{1-2 x}}{5 x+3}+\frac{931 \sqrt{1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac{6899 \sqrt{1-2 x}}{18 (5 x+3)^2}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-6899*Sqrt[1 - 2*x])/(18*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 +
5*x)^2) + (931*Sqrt[1 - 2*x])/(18*(2 + 3*x)*(3 + 5*x)^2) + (2311*Sqrt[1 - 2*x])
/(3 + 5*x) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14073*Sqrt[11/5]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.318668, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac{2311 \sqrt{1-2 x}}{5 x+3}+\frac{931 \sqrt{1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac{6899 \sqrt{1-2 x}}{18 (5 x+3)^2}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-6899*Sqrt[1 - 2*x])/(18*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 +
5*x)^2) + (931*Sqrt[1 - 2*x])/(18*(2 + 3*x)*(3 + 5*x)^2) + (2311*Sqrt[1 - 2*x])
/(3 + 5*x) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14073*Sqrt[11/5]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.5004, size = 131, normalized size = 0.9 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{2311 \sqrt{- 2 x + 1}}{5 x + 3} - \frac{6899 \sqrt{- 2 x + 1}}{18 \left (5 x + 3\right )^{2}} + \frac{931 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + 4555 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )} - \frac{14073 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

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)**3,x)

[Out]

7*(-2*x + 1)**(3/2)/(6*(3*x + 2)**2*(5*x + 3)**2) + 2311*sqrt(-2*x + 1)/(5*x + 3
) - 6899*sqrt(-2*x + 1)/(18*(5*x + 3)**2) + 931*sqrt(-2*x + 1)/(18*(3*x + 2)*(5*
x + 3)**2) + 4555*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7) - 14073*sqrt(55)*ata
nh(sqrt(55)*sqrt(-2*x + 1)/11)/5

_______________________________________________________________________________________

Mathematica [A]  time = 0.15864, size = 95, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (207990 x^3+395215 x^2+249939 x+52607\right )}{2 (3 x+2)^2 (5 x+3)^2}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(52607 + 249939*x + 395215*x^2 + 207990*x^3))/(2*(2 + 3*x)^2*(3 +
5*x)^2) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14073*Sqrt[11/5]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 94, normalized size = 0.7 \[ -252\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{67\, \left ( 1-2\,x \right ) ^{3/2}}{4}}-{\frac{1421\,\sqrt{1-2\,x}}{36}} \right ) }+4555\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+1100\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{207\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{451\,\sqrt{1-2\,x}}{20}} \right ) }-{\frac{14073\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-252*(67/4*(1-2*x)^(3/2)-1421/36*(1-2*x)^(1/2))/(-4-6*x)^2+4555*arctanh(1/7*21^(
1/2)*(1-2*x)^(1/2))*21^(1/2)+1100*(-207/20*(1-2*x)^(3/2)+451/20*(1-2*x)^(1/2))/(
-6-10*x)^2-14073/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49643, size = 197, normalized size = 1.36 \[ \frac{14073}{10} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4555}{2} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (103995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 707200 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1602293 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1209516 \, \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14073/10*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 4555/2*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 2*(103995*(-2*x + 1)^(7/2) - 707200*(-2*x + 1)^(5/2) + 1602293*(-2*x +
1)^(3/2) - 1209516*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(
2*x - 1)^2 + 20944*x - 4543)

_______________________________________________________________________________________

Fricas [A]  time = 0.2321, size = 221, normalized size = 1.52 \[ \frac{\sqrt{5}{\left (4555 \, \sqrt{21} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 14073 \, \sqrt{11}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (207990 \, x^{3} + 395215 \, x^{2} + 249939 \, x + 52607\right )} \sqrt{-2 \, x + 1}\right )}}{10 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*sqrt(5)*(4555*sqrt(21)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*l
og((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 14073*sqrt(11)*(225*x^4 + 57
0*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1)
)/(5*x + 3)) + sqrt(5)*(207990*x^3 + 395215*x^2 + 249939*x + 52607)*sqrt(-2*x +
1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

_______________________________________________________________________________________

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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213942, size = 200, normalized size = 1.38 \[ \frac{14073}{10} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{4555}{2} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (103995 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 707200 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1602293 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1209516 \, \sqrt{-2 \, x + 1}\right )}}{{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14073/10*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 4555/2*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) + 2*(103995*(2*x - 1)^3*sqrt(-2*x + 1) + 707200*(2*x -
1)^2*sqrt(-2*x + 1) - 1602293*(-2*x + 1)^(3/2) + 1209516*sqrt(-2*x + 1))/(15*(2*
x - 1)^2 + 136*x + 9)^2