3.2084 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac{2245 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x]*(541 + 857*x))
/(2058*(2 + 3*x)^2) + (2245*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.114531, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac{2245 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x]*(541 + 857*x))
/(2058*(2 + 3*x)^2) + (2245*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi in Sympy [A]  time = 12.1769, size = 71, normalized size = 0.89 \[ \frac{\sqrt{- 2 x + 1} \left (12855 x + 8115\right )}{6174 \left (3 x + 2\right )^{2}} + \frac{2245 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} + \frac{11 \left (5 x + 3\right )^{2}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(12855*x + 8115)/(6174*(3*x + 2)**2) + 2245*sqrt(21)*atanh(sqrt(2
1)*sqrt(-2*x + 1)/7)/21609 + 11*(5*x + 3)**2/(7*sqrt(-2*x + 1)*(3*x + 2)**2)

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Mathematica [A]  time = 0.134666, size = 58, normalized size = 0.72 \[ \frac{\frac{21 \left (72280 x^2+95895 x+31811\right )}{\sqrt{1-2 x} (3 x+2)^2}+4490 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{43218} \]

Antiderivative was successfully verified.

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

[Out]

((21*(31811 + 95895*x + 72280*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^2) + 4490*Sqrt[21]*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/43218

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Maple [A]  time = 0.019, size = 57, normalized size = 0.7 \[{\frac{1331}{343}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{18}{343\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{203}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{469}{54}\sqrt{1-2\,x}} \right ) }+{\frac{2245\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/343/(1-2*x)^(1/2)-18/343*(-203/54*(1-2*x)^(3/2)+469/54*(1-2*x)^(1/2))/(-4-6
*x)^2+2245/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49392, size = 112, normalized size = 1.4 \[ -\frac{2245}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (18070 \,{\left (2 \, x - 1\right )}^{2} + 168175 \, x + 13741\right )}}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2245/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 2/1029*(18070*(2*x - 1)^2 + 168175*x + 13741)/(9*(-2*x + 1)^(5/2) - 42*
(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]  time = 0.240312, size = 116, normalized size = 1.45 \[ \frac{\sqrt{21}{\left (2245 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (72280 \, x^{2} + 95895 \, x + 31811\right )}\right )}}{43218 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43218*sqrt(21)*(2245*(9*x^2 + 12*x + 4)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5)
- 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(72280*x^2 + 95895*x + 31811))/((9*x
^2 + 12*x + 4)*sqrt(-2*x + 1))

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.238825, size = 104, normalized size = 1.3 \[ -\frac{2245}{43218} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1331}{343 \, \sqrt{-2 \, x + 1}} + \frac{29 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 67 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2245/43218*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 1331/343/sqrt(-2*x + 1) + 1/588*(29*(-2*x + 1)^(3/2) - 67*sqrt(
-2*x + 1))/(3*x + 2)^2