3.1832 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{10 (5 x+3)^2}-\frac{49 \sqrt{1-2 x} (3 x+2)^2}{275 (5 x+3)}+\frac{21 \sqrt{1-2 x} (75 x+44)}{2750}-\frac{1267 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (49*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(
275*(3 + 5*x)) + (21*Sqrt[1 - 2*x]*(44 + 75*x))/2750 - (1267*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(1375*Sqrt[55])

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Rubi [A]  time = 0.153089, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^3}{10 (5 x+3)^2}-\frac{49 \sqrt{1-2 x} (3 x+2)^2}{275 (5 x+3)}+\frac{21 \sqrt{1-2 x} (75 x+44)}{2750}-\frac{1267 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (49*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(
275*(3 + 5*x)) + (21*Sqrt[1 - 2*x]*(44 + 75*x))/2750 - (1267*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(1375*Sqrt[55])

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Rubi in Sympy [A]  time = 19.0843, size = 85, normalized size = 0.85 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{10 \left (5 x + 3\right )^{2}} - \frac{49 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{275 \left (5 x + 3\right )} + \frac{\sqrt{- 2 x + 1} \left (23625 x + 13860\right )}{41250} - \frac{1267 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{75625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**3/(10*(5*x + 3)**2) - 49*sqrt(-2*x + 1)*(3*x + 2)**2/
(275*(5*x + 3)) + sqrt(-2*x + 1)*(23625*x + 13860)/41250 - 1267*sqrt(55)*atanh(s
qrt(55)*sqrt(-2*x + 1)/11)/75625

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Mathematica [A]  time = 0.102445, size = 63, normalized size = 0.63 \[ \frac{\frac{55 \sqrt{1-2 x} \left (9900 x^3+12870 x^2+4555 x+236\right )}{(5 x+3)^2}-2534 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{151250} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(236 + 4555*x + 12870*x^2 + 9900*x^3))/(3 + 5*x)^2 - 2534*Sqr
t[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/151250

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[ -{\frac{9}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{54}{625}\sqrt{1-2\,x}}+{\frac{2}{25\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{197}{110} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{199}{50}\sqrt{1-2\,x}} \right ) }-{\frac{1267\,\sqrt{55}}{75625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/125*(1-2*x)^(3/2)+54/625*(1-2*x)^(1/2)+2/25*(197/110*(1-2*x)^(3/2)-199/50*(1-
2*x)^(1/2))/(-6-10*x)^2-1267/75625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49875, size = 124, normalized size = 1.24 \[ -\frac{9}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1267}{151250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{54}{625} \, \sqrt{-2 \, x + 1} + \frac{985 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2189 \, \sqrt{-2 \, x + 1}}{6875 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/125*(-2*x + 1)^(3/2) + 1267/151250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1)
)/(sqrt(55) + 5*sqrt(-2*x + 1))) + 54/625*sqrt(-2*x + 1) + 1/6875*(985*(-2*x + 1
)^(3/2) - 2189*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.214679, size = 113, normalized size = 1.13 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (9900 \, x^{3} + 12870 \, x^{2} + 4555 \, x + 236\right )} \sqrt{-2 \, x + 1} + 1267 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{151250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/151250*sqrt(55)*(sqrt(55)*(9900*x^3 + 12870*x^2 + 4555*x + 236)*sqrt(-2*x + 1)
+ 1267*(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)

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Sympy [A]  time = 152.922, size = 335, normalized size = 3.35 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{3}{2}}}{125} + \frac{54 \sqrt{- 2 x + 1}}{625} - \frac{388 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{625} + \frac{88 \left (\begin{cases} \frac{\sqrt{55} \left (\frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )^{2}}\right )}{6655} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{625} + \frac{558 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(-2*x + 1)**(3/2)/125 + 54*sqrt(-2*x + 1)/625 - 388*Piecewise((sqrt(55)*(-log
(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(
4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/60
5, (x <= 1/2) & (x > -3/5)))/625 + 88*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(-
2*x + 1)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/16 + 3/(16*(sqrt(55)
*sqrt(-2*x + 1)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)**2) + 3/(16*(s
qrt(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)**2))/66
55, (x <= 1/2) & (x > -3/5)))/625 + 558*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt
(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11
)/55, -2*x + 1 < 11/5))/625

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GIAC/XCAS [A]  time = 0.219212, size = 116, normalized size = 1.16 \[ -\frac{9}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1267}{151250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{54}{625} \, \sqrt{-2 \, x + 1} + \frac{985 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2189 \, \sqrt{-2 \, x + 1}}{27500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/125*(-2*x + 1)^(3/2) + 1267/151250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(
-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 54/625*sqrt(-2*x + 1) + 1/27500*(985
*(-2*x + 1)^(3/2) - 2189*sqrt(-2*x + 1))/(5*x + 3)^2