3.1785 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(1 - 2*x)^(3/2)/(105*(2 + 3*x)^5) - (2*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (2*Sqrt
[1 - 2*x])/(315*(2 + 3*x)^3) + Sqrt[1 - 2*x]/(441*(2 + 3*x)^2) + Sqrt[1 - 2*x]/(
1029*(2 + 3*x)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.125811, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac{\sqrt{1-2 x}}{1029 (3 x+2)}+\frac{\sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{2 \sqrt{1-2 x}}{315 (3 x+2)^3}-\frac{2 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

(1 - 2*x)^(3/2)/(105*(2 + 3*x)^5) - (2*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (2*Sqrt
[1 - 2*x])/(315*(2 + 3*x)^3) + Sqrt[1 - 2*x]/(441*(2 + 3*x)^2) + Sqrt[1 - 2*x]/(
1029*(2 + 3*x)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi in Sympy [A]  time = 13.1683, size = 109, normalized size = 0.85 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{105 \left (3 x + 2\right )^{5}} + \frac{\sqrt{- 2 x + 1}}{1029 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1}}{441 \left (3 x + 2\right )^{2}} + \frac{2 \sqrt{- 2 x + 1}}{315 \left (3 x + 2\right )^{3}} - \frac{2 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{4}} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

(-2*x + 1)**(3/2)/(105*(3*x + 2)**5) + sqrt(-2*x + 1)/(1029*(3*x + 2)) + sqrt(-2
*x + 1)/(441*(3*x + 2)**2) + 2*sqrt(-2*x + 1)/(315*(3*x + 2)**3) - 2*sqrt(-2*x +
1)/(15*(3*x + 2)**4) + 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21609

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Mathematica [A]  time = 0.100768, size = 68, normalized size = 0.53 \[ \frac{\frac{21 \sqrt{1-2 x} \left (405 x^4+1395 x^3+2004 x^2-864 x-1019\right )}{(3 x+2)^5}+10 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{108045} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

((21*Sqrt[1 - 2*x]*(-1019 - 864*x + 2004*x^2 + 1395*x^3 + 405*x^4))/(2 + 3*x)^5
+ 10*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/108045

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ 7776\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{ \left ( 1-2\,x \right ) ^{9/2}}{49392}}+{\frac{ \left ( 1-2\,x \right ) ^{7/2}}{4536}}-{\frac{8\, \left ( 1-2\,x \right ) ^{5/2}}{8505}}+{\frac{13\, \left ( 1-2\,x \right ) ^{3/2}}{13608}}+{\frac{7\,\sqrt{1-2\,x}}{11664}} \right ) }+{\frac{2\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)*(1-2*x)^(1/2)/(2+3*x)^6,x)

[Out]

7776*(-1/49392*(1-2*x)^(9/2)+1/4536*(1-2*x)^(7/2)-8/8505*(1-2*x)^(5/2)+13/13608*
(1-2*x)^(3/2)+7/11664*(1-2*x)^(1/2))/(-4-6*x)^5+2/21609*arctanh(1/7*21^(1/2)*(1-
2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.47698, size = 173, normalized size = 1.35 \[ -\frac{1}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (405 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4410 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 18816 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}\right )}}{5145 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

-1/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 2/5145*(405*(-2*x + 1)^(9/2) - 4410*(-2*x + 1)^(7/2) + 18816*(-2*x + 1)^(5
/2) - 19110*(-2*x + 1)^(3/2) - 12005*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*
x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 0.213214, size = 161, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (405 \, x^{4} + 1395 \, x^{3} + 2004 \, x^{2} - 864 \, x - 1019\right )} \sqrt{-2 \, x + 1} + 5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{108045 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/108045*sqrt(21)*(sqrt(21)*(405*x^4 + 1395*x^3 + 2004*x^2 - 864*x - 1019)*sqrt(
-2*x + 1) + 5*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((sqrt(21
)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 810*x^4 + 1080*x^3 + 720
*x^2 + 240*x + 32)

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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((3+5*x)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219379, size = 157, normalized size = 1.23 \[ -\frac{1}{21609} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{405 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 4410 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 18816 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 19110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12005 \, \sqrt{-2 \, x + 1}}{82320 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="giac")

[Out]

-1/21609*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/82320*(405*(2*x - 1)^4*sqrt(-2*x + 1) + 4410*(2*x - 1)^3*sqrt(-2
*x + 1) + 18816*(2*x - 1)^2*sqrt(-2*x + 1) - 19110*(-2*x + 1)^(3/2) - 12005*sqrt
(-2*x + 1))/(3*x + 2)^5