3.1834 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{(1-2 x)^{3/2}}{110 (5 x+3)^2}-\frac{67 \sqrt{1-2 x}}{550 (5 x+3)}+\frac{67 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

[Out]

-(1 - 2*x)^(3/2)/(110*(3 + 5*x)^2) - (67*Sqrt[1 - 2*x])/(550*(3 + 5*x)) + (67*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi [A]  time = 0.0615903, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(1-2 x)^{3/2}}{110 (5 x+3)^2}-\frac{67 \sqrt{1-2 x}}{550 (5 x+3)}+\frac{67 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(3/2)/(110*(3 + 5*x)^2) - (67*Sqrt[1 - 2*x])/(550*(3 + 5*x)) + (67*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(275*Sqrt[55])

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Rubi in Sympy [A]  time = 7.49919, size = 56, normalized size = 0.82 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{110 \left (5 x + 3\right )^{2}} - \frac{67 \sqrt{- 2 x + 1}}{550 \left (5 x + 3\right )} + \frac{67 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)/(110*(5*x + 3)**2) - 67*sqrt(-2*x + 1)/(550*(5*x + 3)) + 67*s
qrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/15125

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Mathematica [A]  time = 0.0870639, size = 53, normalized size = 0.78 \[ \frac{67 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}}-\frac{\sqrt{1-2 x} (325 x+206)}{550 (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(206 + 325*x))/(550*(3 + 5*x)^2) + (67*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/(275*Sqrt[55])

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Maple [A]  time = 0.016, size = 48, normalized size = 0.7 \[ -100\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{13\, \left ( 1-2\,x \right ) ^{3/2}}{1100}}+{\frac{67\,\sqrt{1-2\,x}}{2500}} \right ) }+{\frac{67\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-100*(-13/1100*(1-2*x)^(3/2)+67/2500*(1-2*x)^(1/2))/(-6-10*x)^2+67/15125*arctanh
(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49686, size = 100, normalized size = 1.47 \[ -\frac{67}{30250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{275 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/30250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 1/275*(325*(-2*x + 1)^(3/2) - 737*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x
+ 11)

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Fricas [A]  time = 0.213425, size = 100, normalized size = 1.47 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (325 \, x + 206\right )} \sqrt{-2 \, x + 1} - 67 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} - 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{30250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30250*sqrt(55)*(sqrt(55)*(325*x + 206)*sqrt(-2*x + 1) - 67*(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 = 92.9142, size = 311, normalized size = 4.57 \[ - \frac{124 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{25} + \frac{88 \left (\begin{cases} \frac{\sqrt{55} \left (\frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )^{2}}\right )}{6655} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{25} - \frac{12 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-124*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(5
5)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5)))/25 + 88*Piecewise((sq
rt(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*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*s
qrt(-2*x + 1)/11 - 1)**2))/6655, (x <= 1/2) & (x > -3/5)))/25 - 12*Piecewise((-s
qrt(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))/25

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GIAC/XCAS [A]  time = 0.217281, size = 92, normalized size = 1.35 \[ -\frac{67}{30250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{325 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 737 \, \sqrt{-2 \, x + 1}}{1100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67/30250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) + 1/1100*(325*(-2*x + 1)^(3/2) - 737*sqrt(-2*x + 1))/(5*x + 3)^2