3.1934 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=121 \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

[Out]

(24965*Sqrt[1 - 2*x])/15876 - (1 - 2*x)^(7/2)/(252*(2 + 3*x)^4) + (31*(1 - 2*x)^
(7/2))/(588*(2 + 3*x)^3) - (4993*(1 - 2*x)^(5/2))/(10584*(2 + 3*x)^2) + (24965*(
1 - 2*x)^(3/2))/(31752*(2 + 3*x)) - (24965*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(22
68*Sqrt[21])

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Rubi [A]  time = 0.143712, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{31 (1-2 x)^{7/2}}{588 (3 x+2)^3}-\frac{(1-2 x)^{7/2}}{252 (3 x+2)^4}-\frac{4993 (1-2 x)^{5/2}}{10584 (3 x+2)^2}+\frac{24965 (1-2 x)^{3/2}}{31752 (3 x+2)}+\frac{24965 \sqrt{1-2 x}}{15876}-\frac{24965 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2268 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(24965*Sqrt[1 - 2*x])/15876 - (1 - 2*x)^(7/2)/(252*(2 + 3*x)^4) + (31*(1 - 2*x)^
(7/2))/(588*(2 + 3*x)^3) - (4993*(1 - 2*x)^(5/2))/(10584*(2 + 3*x)^2) + (24965*(
1 - 2*x)^(3/2))/(31752*(2 + 3*x)) - (24965*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(22
68*Sqrt[21])

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Rubi in Sympy [A]  time = 13.8666, size = 105, normalized size = 0.87 \[ \frac{31 \left (- 2 x + 1\right )^{\frac{7}{2}}}{588 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{252 \left (3 x + 2\right )^{4}} - \frac{4993 \left (- 2 x + 1\right )^{\frac{5}{2}}}{10584 \left (3 x + 2\right )^{2}} + \frac{24965 \left (- 2 x + 1\right )^{\frac{3}{2}}}{31752 \left (3 x + 2\right )} + \frac{24965 \sqrt{- 2 x + 1}}{15876} - \frac{24965 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{47628} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

31*(-2*x + 1)**(7/2)/(588*(3*x + 2)**3) - (-2*x + 1)**(7/2)/(252*(3*x + 2)**4) -
4993*(-2*x + 1)**(5/2)/(10584*(3*x + 2)**2) + 24965*(-2*x + 1)**(3/2)/(31752*(3
*x + 2)) + 24965*sqrt(-2*x + 1)/15876 - 24965*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x
+ 1)/7)/47628

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Mathematica [A]  time = 0.131967, size = 68, normalized size = 0.56 \[ \frac{\frac{21 \sqrt{1-2 x} \left (302400 x^4+1231065 x^3+1526937 x^2+762598 x+134558\right )}{(3 x+2)^4}-49930 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{95256} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(134558 + 762598*x + 1526937*x^2 + 1231065*x^3 + 302400*x^4))
/(2 + 3*x)^4 - 49930*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/95256

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Maple [A]  time = 0.019, size = 75, normalized size = 0.6 \[{\frac{200}{243}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{47185}{672} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{129289}{288} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{824705}{864} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1749055}{2592}\sqrt{1-2\,x}} \right ) }-{\frac{24965\,\sqrt{21}}{47628}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

200/243*(1-2*x)^(1/2)+8/3*(-47185/672*(1-2*x)^(7/2)+129289/288*(1-2*x)^(5/2)-824
705/864*(1-2*x)^(3/2)+1749055/2592*(1-2*x)^(1/2))/(-4-6*x)^4-24965/47628*arctanh
(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50568, size = 161, normalized size = 1.33 \[ \frac{24965}{95256} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{200}{243} \, \sqrt{-2 \, x + 1} - \frac{1273995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 8145207 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12243385 \, \sqrt{-2 \, x + 1}}{6804 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

24965/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 200/243*sqrt(-2*x + 1) - 1/6804*(1273995*(-2*x + 1)^(7/2) - 8145207*(-2
*x + 1)^(5/2) + 17318805*(-2*x + 1)^(3/2) - 12243385*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 = 0.215477, size = 147, normalized size = 1.21 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (302400 \, x^{4} + 1231065 \, x^{3} + 1526937 \, x^{2} + 762598 \, x + 134558\right )} \sqrt{-2 \, x + 1} + 24965 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{95256 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/95256*sqrt(21)*(sqrt(21)*(302400*x^4 + 1231065*x^3 + 1526937*x^2 + 762598*x +
134558)*sqrt(-2*x + 1) + 24965*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqr
t(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 9
6*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/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215345, size = 147, normalized size = 1.21 \[ \frac{24965}{95256} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{200}{243} \, \sqrt{-2 \, x + 1} + \frac{1273995 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 8145207 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 17318805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12243385 \, \sqrt{-2 \, x + 1}}{108864 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

24965/95256*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 200/243*sqrt(-2*x + 1) + 1/108864*(1273995*(2*x - 1)^3*sqrt(-2*
x + 1) + 8145207*(2*x - 1)^2*sqrt(-2*x + 1) - 17318805*(-2*x + 1)^(3/2) + 122433
85*sqrt(-2*x + 1))/(3*x + 2)^4