3.1902 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{129 \sqrt{1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{2643 \sqrt{1-2 x} (3 x+2)^3}{1750}+\frac{1404 \sqrt{1-2 x} (3 x+2)^2}{3125}+\frac{9 \sqrt{1-2 x} (1375 x+32)}{31250}-\frac{12279 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]

[Out]

(1404*Sqrt[1 - 2*x]*(2 + 3*x)^2)/3125 + (2643*Sqrt[1 - 2*x]*(2 + 3*x)^3)/1750 -
((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(10*(3 + 5*x)^2) - (129*Sqrt[1 - 2*x]*(2 + 3*x)^4)
/(50*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(32 + 1375*x))/31250 - (12279*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(15625*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.25783, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{129 \sqrt{1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{2643 \sqrt{1-2 x} (3 x+2)^3}{1750}+\frac{1404 \sqrt{1-2 x} (3 x+2)^2}{3125}+\frac{9 \sqrt{1-2 x} (1375 x+32)}{31250}-\frac{12279 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

(1404*Sqrt[1 - 2*x]*(2 + 3*x)^2)/3125 + (2643*Sqrt[1 - 2*x]*(2 + 3*x)^3)/1750 -
((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(10*(3 + 5*x)^2) - (129*Sqrt[1 - 2*x]*(2 + 3*x)^4)
/(50*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(32 + 1375*x))/31250 - (12279*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(15625*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.6367, size = 116, normalized size = 0.83 \[ - \frac{\left (- 360045 x + 769230\right ) \left (- 2 x + 1\right )^{\frac{3}{2}}}{7218750} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}}{10 \left (5 x + 3\right )^{2}} - \frac{129 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{550 \left (5 x + 3\right )} + \frac{1899 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{9625} + \frac{12279 \sqrt{- 2 x + 1}}{171875} - \frac{12279 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{859375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x)**3,x)

[Out]

-(-360045*x + 769230)*(-2*x + 1)**(3/2)/7218750 - (-2*x + 1)**(3/2)*(3*x + 2)**4
/(10*(5*x + 3)**2) - 129*(-2*x + 1)**(3/2)*(3*x + 2)**3/(550*(5*x + 3)) + 1899*(
-2*x + 1)**(3/2)*(3*x + 2)**2/9625 + 12279*sqrt(-2*x + 1)/171875 - 12279*sqrt(55
)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/859375

_______________________________________________________________________________________

Mathematica [A]  time = 0.132871, size = 73, normalized size = 0.52 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (2025000 x^5+3267000 x^4-496350 x^3-2120880 x^2-489445 x+96776\right )}{(5 x+3)^2}-171906 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12031250} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

((-55*Sqrt[1 - 2*x]*(96776 - 489445*x - 2120880*x^2 - 496350*x^3 + 3267000*x^4 +
2025000*x^5))/(3 + 5*x)^2 - 171906*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/
12031250

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 84, normalized size = 0.6 \[{\frac{81}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{1107}{6250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{36}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{228}{3125}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{259}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2871}{500}\sqrt{1-2\,x}} \right ) }-{\frac{12279\,\sqrt{55}}{859375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^3,x)

[Out]

81/1750*(1-2*x)^(7/2)-1107/6250*(1-2*x)^(5/2)+36/3125*(1-2*x)^(3/2)+228/3125*(1-
2*x)^(1/2)+4/125*(259/100*(1-2*x)^(3/2)-2871/500*(1-2*x)^(1/2))/(-6-10*x)^2-1227
9/859375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.5049, size = 149, normalized size = 1.06 \[ \frac{81}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1107}{6250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{36}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12279}{1718750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{228}{3125} \, \sqrt{-2 \, x + 1} + \frac{1295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2871 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^3,x, algorithm="maxima")

[Out]

81/1750*(-2*x + 1)^(7/2) - 1107/6250*(-2*x + 1)^(5/2) + 36/3125*(-2*x + 1)^(3/2)
+ 12279/1718750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 228/3125*sqrt(-2*x + 1) + 1/15625*(1295*(-2*x + 1)^(3/2) - 2871*sq
rt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

_______________________________________________________________________________________

Fricas [A]  time = 0.215846, size = 127, normalized size = 0.91 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (2025000 \, x^{5} + 3267000 \, x^{4} - 496350 \, x^{3} - 2120880 \, x^{2} - 489445 \, x + 96776\right )} \sqrt{-2 \, x + 1} - 85953 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{12031250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^3,x, algorithm="fricas")

[Out]

-1/12031250*sqrt(55)*(sqrt(55)*(2025000*x^5 + 3267000*x^4 - 496350*x^3 - 2120880
*x^2 - 489445*x + 96776)*sqrt(-2*x + 1) - 85953*(25*x^2 + 30*x + 9)*log((sqrt(55
)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2 + 30*x + 9)

_______________________________________________________________________________________

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)**(3/2)*(2+3*x)**4/(3+5*x)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216504, size = 159, normalized size = 1.14 \[ -\frac{81}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1107}{6250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{36}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12279}{1718750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{228}{3125} \, \sqrt{-2 \, x + 1} + \frac{1295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2871 \, \sqrt{-2 \, x + 1}}{62500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(3/2)/(5*x + 3)^3,x, algorithm="giac")

[Out]

-81/1750*(2*x - 1)^3*sqrt(-2*x + 1) - 1107/6250*(2*x - 1)^2*sqrt(-2*x + 1) + 36/
3125*(-2*x + 1)^(3/2) + 12279/1718750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(
-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 228/3125*sqrt(-2*x + 1) + 1/62500*(1
295*(-2*x + 1)^(3/2) - 2871*sqrt(-2*x + 1))/(5*x + 3)^2