3.2148 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{720 x+487}{294 \sqrt{1-2 x} (3 x+2)^2}+\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

[Out]

(905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 +
3*x)^2) - (487 + 720*x)/(294*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (905*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi [A]  time = 0.141183, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{720 x+487}{294 \sqrt{1-2 x} (3 x+2)^2}+\frac{905 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{905 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(905*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 +
3*x)^2) - (487 + 720*x)/(294*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (905*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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Rubi in Sympy [A]  time = 14.0894, size = 87, normalized size = 0.87 \[ \frac{905 \sqrt{- 2 x + 1}}{2058 \left (3 x + 2\right )} + \frac{905 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} - \frac{15120 x + 10227}{6174 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{11 \left (5 x + 3\right )^{2}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

905*sqrt(-2*x + 1)/(2058*(3*x + 2)) + 905*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)
/7)/21609 - (15120*x + 10227)/(6174*sqrt(-2*x + 1)*(3*x + 2)**2) + 11*(5*x + 3)*
*2/(21*(-2*x + 1)**(3/2)*(3*x + 2)**2)

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Mathematica [A]  time = 0.131074, size = 66, normalized size = 0.66 \[ \frac{\frac{21 \sqrt{1-2 x} \left (10860 x^3+33410 x^2+29593 x+8103\right )}{\left (6 x^2+x-2\right )^2}+1810 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{43218} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(8103 + 29593*x + 33410*x^2 + 10860*x^3))/(-2 + x + 6*x^2)^2
+ 1810*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/43218

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Maple [A]  time = 0.021, size = 66, normalized size = 0.7 \[{\frac{1331}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{726}{2401}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{18}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{199}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1379}{54}\sqrt{1-2\,x}} \right ) }+{\frac{905\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1331/1029/(1-2*x)^(3/2)-726/2401/(1-2*x)^(1/2)-18/2401*(-199/18*(1-2*x)^(3/2)+13
79/54*(1-2*x)^(1/2))/(-4-6*x)^2+905/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21
^(1/2)

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Maxima [A]  time = 1.51275, size = 124, normalized size = 1.24 \[ -\frac{905}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2715 \,{\left (2 \, x - 1\right )}^{3} + 24850 \,{\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/1029*(2715*(2*x - 1)^3 + 24850*(2*x - 1)^2 + 142296*x - 5929)/(9*(-2*x
+ 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.210863, size = 138, normalized size = 1.38 \[ \frac{\sqrt{21}{\left (905 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{21}{\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )}\right )}}{43218 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43218*sqrt(21)*(905*(18*x^3 + 15*x^2 - 4*x - 4)*sqrt(-2*x + 1)*log((sqrt(21)*(
3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(21)*(10860*x^3 + 33410*x^2 + 295
93*x + 8103))/((18*x^3 + 15*x^2 - 4*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)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.229446, size = 120, normalized size = 1.2 \[ -\frac{905}{43218} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{121 \,{\left (36 \, x + 59\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{597 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1379 \, \sqrt{-2 \, x + 1}}{28812 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-905/43218*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 121/7203*(36*x + 59)/((2*x - 1)*sqrt(-2*x + 1)) + 1/28812*(597*(
-2*x + 1)^(3/2) - 1379*sqrt(-2*x + 1))/(3*x + 2)^2