3.1976 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{129 (1-2 x)^{7/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{7/2}}{550 (5 x+3)^2}+\frac{1533 (1-2 x)^{5/2}}{75625}+\frac{511 (1-2 x)^{3/2}}{6875}+\frac{1533 \sqrt{1-2 x}}{3125}-\frac{1533 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(1533*Sqrt[1 - 2*x])/3125 + (511*(1 - 2*x)^(3/2))/6875 + (1533*(1 - 2*x)^(5/2))/
75625 - (1 - 2*x)^(7/2)/(550*(3 + 5*x)^2) - (129*(1 - 2*x)^(7/2))/(6050*(3 + 5*x
)) - (1533*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.14179, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{129 (1-2 x)^{7/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{7/2}}{550 (5 x+3)^2}+\frac{1533 (1-2 x)^{5/2}}{75625}+\frac{511 (1-2 x)^{3/2}}{6875}+\frac{1533 \sqrt{1-2 x}}{3125}-\frac{1533 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(1533*Sqrt[1 - 2*x])/3125 + (511*(1 - 2*x)^(3/2))/6875 + (1533*(1 - 2*x)^(5/2))/
75625 - (1 - 2*x)^(7/2)/(550*(3 + 5*x)^2) - (129*(1 - 2*x)^(7/2))/(6050*(3 + 5*x
)) - (1533*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 12.5627, size = 92, normalized size = 0.84 \[ - \frac{129 \left (- 2 x + 1\right )^{\frac{7}{2}}}{6050 \left (5 x + 3\right )} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{550 \left (5 x + 3\right )^{2}} + \frac{1533 \left (- 2 x + 1\right )^{\frac{5}{2}}}{75625} + \frac{511 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6875} + \frac{1533 \sqrt{- 2 x + 1}}{3125} - \frac{1533 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

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]

-129*(-2*x + 1)**(7/2)/(6050*(5*x + 3)) - (-2*x + 1)**(7/2)/(550*(5*x + 3)**2) +
1533*(-2*x + 1)**(5/2)/75625 + 511*(-2*x + 1)**(3/2)/6875 + 1533*sqrt(-2*x + 1)
/3125 - 1533*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.11946, size = 68, normalized size = 0.62 \[ \frac{\frac{5 \sqrt{1-2 x} \left (18000 x^4-25400 x^3+51980 x^2+98595 x+32504\right )}{(5 x+3)^2}-3066 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{31250} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(32504 + 98595*x + 51980*x^2 - 25400*x^3 + 18000*x^4))/(3 + 5*
x)^2 - 3066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/31250

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Maple [A]  time = 0.017, size = 75, normalized size = 0.7 \[{\frac{18}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{58}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1658}{3125}\sqrt{1-2\,x}}+{\frac{22}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{123}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{55}{2}\sqrt{1-2\,x}} \right ) }-{\frac{1533\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

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]

18/625*(1-2*x)^(5/2)+58/625*(1-2*x)^(3/2)+1658/3125*(1-2*x)^(1/2)+22/125*(123/10
*(1-2*x)^(3/2)-55/2*(1-2*x)^(1/2))/(-6-10*x)^2-1533/15625*arctanh(1/11*55^(1/2)*
(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49904, size = 136, normalized size = 1.25 \[ \frac{18}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{58}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1533}{31250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1658}{3125} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (123 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 275 \, \sqrt{-2 \, x + 1}\right )}}{625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/625*(-2*x + 1)^(5/2) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*sqrt(55)*log(-(sq
rt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1658/3125*sqrt(-2*x
+ 1) + 11/625*(123*(-2*x + 1)^(3/2) - 275*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*
x + 11)

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Fricas [A]  time = 0.221521, size = 128, normalized size = 1.17 \[ \frac{\sqrt{5}{\left (1533 \, \sqrt{11}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (18000 \, x^{4} - 25400 \, x^{3} + 51980 \, x^{2} + 98595 \, x + 32504\right )} \sqrt{-2 \, x + 1}\right )}}{31250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/31250*sqrt(5)*(1533*sqrt(11)*(25*x^2 + 30*x + 9)*log((sqrt(5)*(5*x - 8) + 5*sq
rt(11)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(18000*x^4 - 25400*x^3 + 51980*x^2 +
98595*x + 32504)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213798, size = 138, normalized size = 1.27 \[ \frac{18}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{58}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1533}{31250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1658}{3125} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (123 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 275 \, \sqrt{-2 \, x + 1}\right )}}{2500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18/625*(2*x - 1)^2*sqrt(-2*x + 1) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*sqrt(55
)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1
658/3125*sqrt(-2*x + 1) + 11/2500*(123*(-2*x + 1)^(3/2) - 275*sqrt(-2*x + 1))/(5
*x + 3)^2