3.1922 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ \frac{(1-2 x)^{7/2}}{84 (3 x+2)^4}-\frac{139 (1-2 x)^{5/2}}{756 (3 x+2)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (3 x+2)^2}-\frac{695 \sqrt{1-2 x}}{4536 (3 x+2)}+\frac{695 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

[Out]

(1 - 2*x)^(7/2)/(84*(2 + 3*x)^4) - (139*(1 - 2*x)^(5/2))/(756*(2 + 3*x)^3) + (69
5*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^2) - (695*Sqrt[1 - 2*x])/(4536*(2 + 3*x)) + (
695*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

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Rubi [A]  time = 0.109508, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(1-2 x)^{7/2}}{84 (3 x+2)^4}-\frac{139 (1-2 x)^{5/2}}{756 (3 x+2)^3}+\frac{695 (1-2 x)^{3/2}}{4536 (3 x+2)^2}-\frac{695 \sqrt{1-2 x}}{4536 (3 x+2)}+\frac{695 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(1 - 2*x)^(7/2)/(84*(2 + 3*x)^4) - (139*(1 - 2*x)^(5/2))/(756*(2 + 3*x)^3) + (69
5*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^2) - (695*Sqrt[1 - 2*x])/(4536*(2 + 3*x)) + (
695*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(2268*Sqrt[21])

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Rubi in Sympy [A]  time = 11.4768, size = 94, normalized size = 0.87 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{84 \left (3 x + 2\right )^{4}} - \frac{139 \left (- 2 x + 1\right )^{\frac{5}{2}}}{756 \left (3 x + 2\right )^{3}} + \frac{695 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4536 \left (3 x + 2\right )^{2}} - \frac{695 \sqrt{- 2 x + 1}}{4536 \left (3 x + 2\right )} + \frac{695 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{47628} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(7/2)/(84*(3*x + 2)**4) - 139*(-2*x + 1)**(5/2)/(756*(3*x + 2)**3) +
695*(-2*x + 1)**(3/2)/(4536*(3*x + 2)**2) - 695*sqrt(-2*x + 1)/(4536*(3*x + 2))
+ 695*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/47628

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Mathematica [A]  time = 0.113138, size = 63, normalized size = 0.58 \[ \frac{1390 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} \left (41715 x^3+43971 x^2+18394 x+4394\right )}{(3 x+2)^4}}{95256} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(4394 + 18394*x + 43971*x^2 + 41715*x^3))/(2 + 3*x)^4 + 1390
*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/95256

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Maple [A]  time = 0.017, size = 66, normalized size = 0.6 \[ -1296\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{515\, \left ( 1-2\,x \right ) ^{7/2}}{36288}}+{\frac{10147\, \left ( 1-2\,x \right ) ^{5/2}}{139968}}-{\frac{53515\, \left ( 1-2\,x \right ) ^{3/2}}{419904}}+{\frac{34055\,\sqrt{1-2\,x}}{419904}} \right ) }+{\frac{695\,\sqrt{21}}{47628}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1296*(-515/36288*(1-2*x)^(7/2)+10147/139968*(1-2*x)^(5/2)-53515/419904*(1-2*x)^
(3/2)+34055/419904*(1-2*x)^(1/2))/(-4-6*x)^4+695/47628*arctanh(1/7*21^(1/2)*(1-2
*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.56576, size = 149, normalized size = 1.38 \[ -\frac{695}{95256} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{41715 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 213087 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 374605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 238385 \, \sqrt{-2 \, x + 1}}{2268 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-695/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/2268*(41715*(-2*x + 1)^(7/2) - 213087*(-2*x + 1)^(5/2) + 374605*(-2*x
+ 1)^(3/2) - 238385*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*
x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.21496, size = 140, normalized size = 1.3 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (41715 \, x^{3} + 43971 \, x^{2} + 18394 \, x + 4394\right )} \sqrt{-2 \, x + 1} - 695 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{95256 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/95256*sqrt(21)*(sqrt(21)*(41715*x^3 + 43971*x^2 + 18394*x + 4394)*sqrt(-2*x +
1) - 695*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) - 21*
sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21351, size = 135, normalized size = 1.25 \[ -\frac{695}{95256} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{41715 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 213087 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 374605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 238385 \, \sqrt{-2 \, x + 1}}{36288 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-695/95256*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 1/36288*(41715*(2*x - 1)^3*sqrt(-2*x + 1) + 213087*(2*x - 1)^2*s
qrt(-2*x + 1) - 374605*(-2*x + 1)^(3/2) + 238385*sqrt(-2*x + 1))/(3*x + 2)^4