3.1783 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ \frac{(1-2 x)^{3/2}}{63 (3 x+2)^3}+\frac{17 \sqrt{1-2 x}}{441 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{63 (3 x+2)^2}+\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{441 \sqrt{21}} \]

[Out]

(1 - 2*x)^(3/2)/(63*(2 + 3*x)^3) - (17*Sqrt[1 - 2*x])/(63*(2 + 3*x)^2) + (17*Sqr
t[1 - 2*x])/(441*(2 + 3*x)) + (34*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21
])

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Rubi [A]  time = 0.079383, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{3/2}}{63 (3 x+2)^3}+\frac{17 \sqrt{1-2 x}}{441 (3 x+2)}-\frac{17 \sqrt{1-2 x}}{63 (3 x+2)^2}+\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{441 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(1 - 2*x)^(3/2)/(63*(2 + 3*x)^3) - (17*Sqrt[1 - 2*x])/(63*(2 + 3*x)^2) + (17*Sqr
t[1 - 2*x])/(441*(2 + 3*x)) + (34*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(441*Sqrt[21
])

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Rubi in Sympy [A]  time = 9.8744, size = 75, normalized size = 0.85 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{63 \left (3 x + 2\right )^{3}} + \frac{17 \sqrt{- 2 x + 1}}{441 \left (3 x + 2\right )} - \frac{17 \sqrt{- 2 x + 1}}{63 \left (3 x + 2\right )^{2}} + \frac{34 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9261} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)/(63*(3*x + 2)**3) + 17*sqrt(-2*x + 1)/(441*(3*x + 2)) - 17*sqr
t(-2*x + 1)/(63*(3*x + 2)**2) + 34*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/926
1

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Mathematica [A]  time = 0.0895572, size = 58, normalized size = 0.66 \[ \frac{\frac{21 \sqrt{1-2 x} \left (153 x^2-167 x-163\right )}{(3 x+2)^3}+34 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-163 - 167*x + 153*x^2))/(2 + 3*x)^3 + 34*Sqrt[21]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/9261

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Maple [A]  time = 0.017, size = 57, normalized size = 0.7 \[ 216\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{17\, \left ( 1-2\,x \right ) ^{5/2}}{5292}}-{\frac{ \left ( 1-2\,x \right ) ^{3/2}}{1701}}+{\frac{17\,\sqrt{1-2\,x}}{972}} \right ) }+{\frac{34\,\sqrt{21}}{9261}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

216*(-17/5292*(1-2*x)^(5/2)-1/1701*(1-2*x)^(3/2)+17/972*(1-2*x)^(1/2))/(-4-6*x)^
3+34/9261*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.5161, size = 124, normalized size = 1.41 \[ -\frac{17}{9261} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (153 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 28 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 833 \, \sqrt{-2 \, x + 1}\right )}}{441 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17/9261*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 2/441*(153*(-2*x + 1)^(5/2) + 28*(-2*x + 1)^(3/2) - 833*sqrt(-2*x + 1))/(2
7*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.211377, size = 120, normalized size = 1.36 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (153 \, x^{2} - 167 \, x - 163\right )} \sqrt{-2 \, x + 1} + 17 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{9261 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9261*sqrt(21)*(sqrt(21)*(153*x^2 - 167*x - 163)*sqrt(-2*x + 1) + 17*(27*x^3 +
54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(27*
x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 163.61, size = 439, normalized size = 4.99 \[ \frac{40 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} + \frac{296 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} + \frac{112 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{5 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{32} + \frac{5 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{32} - \frac{5}{32 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} - \frac{1}{48 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{3}} - \frac{5}{32 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}} - \frac{1}{48 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{3}}\right )}{7203} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqr
t(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sq
rt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/9 + 296*Piecewise((sqrt(21)
*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)
/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7
+ 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x +
1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/9 + 112*Piecewise((sqrt(21)*(-5*l
og(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/32 -
5/(32*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)*
*2) - 1/(48*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(-2*x + 1)
/7 - 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(-2*x
+ 1)/7 - 1)**3))/7203, (x <= 1/2) & (x > -2/3)))/9

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GIAC/XCAS [A]  time = 0.213245, size = 113, normalized size = 1.28 \[ -\frac{17}{9261} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{153 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 28 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 833 \, \sqrt{-2 \, x + 1}}{1764 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17/9261*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/1764*(153*(2*x - 1)^2*sqrt(-2*x + 1) + 28*(-2*x + 1)^(3/2) - 833
*sqrt(-2*x + 1))/(3*x + 2)^3