3.2149 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{2 (2027 x+1346)}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{3755 \sqrt{1-2 x}}{7203 (3 x+2)}-\frac{3755 \sqrt{1-2 x}}{3087 (3 x+2)^2}-\frac{7510 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203 \sqrt{21}} \]

[Out]

(-3755*Sqrt[1 - 2*x])/(3087*(2 + 3*x)^2) - (3755*Sqrt[1 - 2*x])/(7203*(2 + 3*x))
+ (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (2*(1346 + 2027*x))/(441*
Sqrt[1 - 2*x]*(2 + 3*x)^3) - (7510*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7203*Sqrt[
21])

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Rubi [A]  time = 0.161502, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{2 (2027 x+1346)}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{3755 \sqrt{1-2 x}}{7203 (3 x+2)}-\frac{3755 \sqrt{1-2 x}}{3087 (3 x+2)^2}-\frac{7510 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7203 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-3755*Sqrt[1 - 2*x])/(3087*(2 + 3*x)^2) - (3755*Sqrt[1 - 2*x])/(7203*(2 + 3*x))
+ (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (2*(1346 + 2027*x))/(441*
Sqrt[1 - 2*x]*(2 + 3*x)^3) - (7510*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7203*Sqrt[
21])

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Rubi in Sympy [A]  time = 16.4618, size = 105, normalized size = 0.88 \[ - \frac{3755 \sqrt{- 2 x + 1}}{7203 \left (3 x + 2\right )} - \frac{3755 \sqrt{- 2 x + 1}}{3087 \left (3 x + 2\right )^{2}} - \frac{7510 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{151263} + \frac{85134 x + 56532}{9261 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{11 \left (5 x + 3\right )^{2}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3755*sqrt(-2*x + 1)/(7203*(3*x + 2)) - 3755*sqrt(-2*x + 1)/(3087*(3*x + 2)**2)
- 7510*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/151263 + (85134*x + 56532)/(926
1*sqrt(-2*x + 1)*(3*x + 2)**3) + 11*(5*x + 3)**2/(21*(-2*x + 1)**(3/2)*(3*x + 2)
**3)

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Mathematica [A]  time = 0.164386, size = 68, normalized size = 0.57 \[ \frac{-\frac{21 \left (135180 x^4+150200 x^3-83306 x^2-150295 x-45383\right )}{(1-2 x)^{3/2} (3 x+2)^3}-7510 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{151263} \]

Antiderivative was successfully verified.

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

[Out]

((-21*(-45383 - 150295*x - 83306*x^2 + 150200*x^3 + 135180*x^4))/((1 - 2*x)^(3/2
)*(2 + 3*x)^3) - 7510*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/151263

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Maple [A]  time = 0.022, size = 75, normalized size = 0.6 \[{\frac{2662}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{6534}{16807}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{54}{16807\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3118}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{128870}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{147980}{81}\sqrt{1-2\,x}} \right ) }-{\frac{7510\,\sqrt{21}}{151263}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2662/7203/(1-2*x)^(3/2)+6534/16807/(1-2*x)^(1/2)+54/16807*(-3118/9*(1-2*x)^(5/2)
+128870/81*(1-2*x)^(3/2)-147980/81*(1-2*x)^(1/2))/(-4-6*x)^3-7510/151263*arctanh
(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.59782, size = 149, normalized size = 1.24 \[ \frac{3755}{151263} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{4} + 210280 \,{\left (2 \, x - 1\right )}^{3} + 344764 \,{\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3755/151263*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 2/7203*(33795*(2*x - 1)^4 + 210280*(2*x - 1)^3 + 344764*(2*x - 1)^2 - 2
13444*x - 349811)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x + 1)^(
5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.221438, size = 157, normalized size = 1.31 \[ \frac{\sqrt{21}{\left (3755 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (135180 \, x^{4} + 150200 \, x^{3} - 83306 \, x^{2} - 150295 \, x - 45383\right )}\right )}}{151263 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/151263*sqrt(21)*(3755*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*sqrt(-2*x + 1)*log
((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(135180*x^4 + 15
0200*x^3 - 83306*x^2 - 150295*x - 45383))/((54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
*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)**(5/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221481, size = 128, normalized size = 1.07 \[ \frac{3755}{151263} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{4} + 210280 \,{\left (2 \, x - 1\right )}^{3} + 344764 \,{\left (2 \, x - 1\right )}^{2} - 213444 \, x - 349811\right )}}{7203 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3755/151263*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 2/7203*(33795*(2*x - 1)^4 + 210280*(2*x - 1)^3 + 344764*(2*x -
1)^2 - 213444*x - 349811)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3