3.1993 \(\int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ -\frac{50 \sqrt{1-2 x}}{1029 (3 x+2)}-\frac{50 \sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{63 (3 x+2)^3}-\frac{100 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

Sqrt[1 - 2*x]/(63*(2 + 3*x)^3) - (50*Sqrt[1 - 2*x])/(441*(2 + 3*x)^2) - (50*Sqrt
[1 - 2*x])/(1029*(2 + 3*x)) - (100*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[
21])

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Rubi [A]  time = 0.090328, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{50 \sqrt{1-2 x}}{1029 (3 x+2)}-\frac{50 \sqrt{1-2 x}}{441 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{63 (3 x+2)^3}-\frac{100 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(63*(2 + 3*x)^3) - (50*Sqrt[1 - 2*x])/(441*(2 + 3*x)^2) - (50*Sqrt
[1 - 2*x])/(1029*(2 + 3*x)) - (100*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[
21])

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Rubi in Sympy [A]  time = 8.72895, size = 75, normalized size = 0.85 \[ - \frac{50 \sqrt{- 2 x + 1}}{1029 \left (3 x + 2\right )} - \frac{50 \sqrt{- 2 x + 1}}{441 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{63 \left (3 x + 2\right )^{3}} - \frac{100 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-50*sqrt(-2*x + 1)/(1029*(3*x + 2)) - 50*sqrt(-2*x + 1)/(441*(3*x + 2)**2) + sqr
t(-2*x + 1)/(63*(3*x + 2)**3) - 100*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21
609

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Mathematica [A]  time = 0.0976841, size = 58, normalized size = 0.66 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (450 x^2+950 x+417\right )}{(3 x+2)^3}-100 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21609} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(417 + 950*x + 450*x^2))/(2 + 3*x)^3 - 100*Sqrt[21]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/21609

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Maple [A]  time = 0.014, size = 57, normalized size = 0.7 \[ 216\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{25\, \left ( 1-2\,x \right ) ^{5/2}}{6174}}-{\frac{100\, \left ( 1-2\,x \right ) ^{3/2}}{3969}}+{\frac{41\,\sqrt{1-2\,x}}{1134}} \right ) }-{\frac{100\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

216*(25/6174*(1-2*x)^(5/2)-100/3969*(1-2*x)^(3/2)+41/1134*(1-2*x)^(1/2))/(-4-6*x
)^3-100/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49497, size = 124, normalized size = 1.41 \[ \frac{50}{21609} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 1400 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2009 \, \sqrt{-2 \, x + 1}\right )}}{1029 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 4/1029*(225*(-2*x + 1)^(5/2) - 1400*(-2*x + 1)^(3/2) + 2009*sqrt(-2*x + 1)
)/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.230433, size = 120, normalized size = 1.36 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (450 \, x^{2} + 950 \, x + 417\right )} \sqrt{-2 \, x + 1} - 50 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{21609 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21609*sqrt(21)*(sqrt(21)*(450*x^2 + 950*x + 417)*sqrt(-2*x + 1) - 50*(27*x^3
+ 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(2
7*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.211222, size = 113, normalized size = 1.28 \[ \frac{50}{21609} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{225 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1400 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2009 \, \sqrt{-2 \, x + 1}}{2058 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

50/21609*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 1/2058*(225*(2*x - 1)^2*sqrt(-2*x + 1) - 1400*(-2*x + 1)^(3/2) + 2
009*sqrt(-2*x + 1))/(3*x + 2)^3