3.2137 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=110 \[ -\frac{255 \sqrt{1-2 x}}{686 (3 x+2)}+\frac{85}{147 \sqrt{1-2 x} (3 x+2)}-\frac{26}{21 \sqrt{1-2 x} (3 x+2)^2}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{85}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - 26/(21*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 85/(1
47*Sqrt[1 - 2*x]*(2 + 3*x)) - (255*Sqrt[1 - 2*x])/(686*(2 + 3*x)) - (85*Sqrt[3/7
]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi [A]  time = 0.137569, antiderivative size = 110, 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 \[ -\frac{255 \sqrt{1-2 x}}{686 (3 x+2)}+\frac{85}{147 \sqrt{1-2 x} (3 x+2)}-\frac{26}{21 \sqrt{1-2 x} (3 x+2)^2}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac{85}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - 26/(21*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 85/(1
47*Sqrt[1 - 2*x]*(2 + 3*x)) - (255*Sqrt[1 - 2*x])/(686*(2 + 3*x)) - (85*Sqrt[3/7
]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343

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Rubi in Sympy [A]  time = 12.3693, size = 94, normalized size = 0.85 \[ - \frac{255 \sqrt{- 2 x + 1}}{686 \left (3 x + 2\right )} - \frac{85 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} + \frac{85}{147 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{26}{21 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{121}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-255*sqrt(-2*x + 1)/(686*(3*x + 2)) - 85*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/
7)/2401 + 85/(147*sqrt(-2*x + 1)*(3*x + 2)) - 26/(21*sqrt(-2*x + 1)*(3*x + 2)**2
) + 121/(42*(-2*x + 1)**(3/2)*(3*x + 2)**2)

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Mathematica [A]  time = 0.132816, size = 66, normalized size = 0.6 \[ \frac{-\frac{7 \sqrt{1-2 x} \left (9180 x^3+4080 x^2-7731 x-4231\right )}{\left (6 x^2+x-2\right )^2}-510 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{14406} \]

Antiderivative was successfully verified.

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

[Out]

((-7*Sqrt[1 - 2*x]*(-4231 - 7731*x + 4080*x^2 + 9180*x^3))/(-2 + x + 6*x^2)^2 -
510*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/14406

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Maple [A]  time = 0.02, size = 66, normalized size = 0.6 \[{\frac{242}{1029} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{638}{2401}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{18}{2401\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{43}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{889}{18}\sqrt{1-2\,x}} \right ) }-{\frac{85\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

242/1029/(1-2*x)^(3/2)+638/2401/(1-2*x)^(1/2)+18/2401*(-43/2*(1-2*x)^(3/2)+889/1
8*(1-2*x)^(1/2))/(-4-6*x)^2-85/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51348, size = 124, normalized size = 1.13 \[ \frac{85}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2295 \,{\left (2 \, x - 1\right )}^{3} + 8925 \,{\left (2 \, x - 1\right )}^{2} + 6468 \, x - 15092}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

85/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 1/1029*(2295*(2*x - 1)^3 + 8925*(2*x - 1)^2 + 6468*x - 15092)/(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 = 0.221709, size = 144, normalized size = 1.31 \[ \frac{\sqrt{7}{\left (255 \, \sqrt{3}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (9180 \, x^{3} + 4080 \, x^{2} - 7731 \, x - 4231\right )}\right )}}{14406 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14406*sqrt(7)*(255*sqrt(3)*(18*x^3 + 15*x^2 - 4*x - 4)*sqrt(-2*x + 1)*log((sqr
t(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(9180*x^3 + 4080
*x^2 - 7731*x - 4231))/((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)**2/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215663, size = 120, normalized size = 1.09 \[ \frac{85}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{44 \,{\left (87 \, x - 82\right )}}{7203 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{387 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 889 \, \sqrt{-2 \, x + 1}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

85/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 44/7203*(87*x - 82)/((2*x - 1)*sqrt(-2*x + 1)) - 1/9604*(387*(-2*x
+ 1)^(3/2) - 889*sqrt(-2*x + 1))/(3*x + 2)^2