3.1795 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ \frac{275 (1-2 x)^{3/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{3/2}}{252 (3 x+2)^4}+\frac{4625 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{4625 \sqrt{1-2 x}}{10584 (3 x+2)^2}+\frac{4625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(3/2)/(252*(2 + 3*x)^4) + (275*(1 - 2*x)^(3/2))/(5292*(2 + 3*x)^3) -
(4625*Sqrt[1 - 2*x])/(10584*(2 + 3*x)^2) + (4625*Sqrt[1 - 2*x])/(74088*(2 + 3*x)
) + (4625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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Rubi [A]  time = 0.1171, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{275 (1-2 x)^{3/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{3/2}}{252 (3 x+2)^4}+\frac{4625 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{4625 \sqrt{1-2 x}}{10584 (3 x+2)^2}+\frac{4625 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(3/2)/(252*(2 + 3*x)^4) + (275*(1 - 2*x)^(3/2))/(5292*(2 + 3*x)^3) -
(4625*Sqrt[1 - 2*x])/(10584*(2 + 3*x)^2) + (4625*Sqrt[1 - 2*x])/(74088*(2 + 3*x)
) + (4625*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(37044*Sqrt[21])

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Rubi in Sympy [A]  time = 12.5388, size = 94, normalized size = 0.87 \[ \frac{275 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5292 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{252 \left (3 x + 2\right )^{4}} + \frac{4625 \sqrt{- 2 x + 1}}{74088 \left (3 x + 2\right )} - \frac{4625 \sqrt{- 2 x + 1}}{10584 \left (3 x + 2\right )^{2}} + \frac{4625 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{777924} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275*(-2*x + 1)**(3/2)/(5292*(3*x + 2)**3) - (-2*x + 1)**(3/2)/(252*(3*x + 2)**4)
+ 4625*sqrt(-2*x + 1)/(74088*(3*x + 2)) - 4625*sqrt(-2*x + 1)/(10584*(3*x + 2)*
*2) + 4625*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/777924

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Mathematica [A]  time = 0.106241, size = 63, normalized size = 0.58 \[ \frac{\frac{21 \sqrt{1-2 x} \left (124875 x^3-64725 x^2-225262 x-85094\right )}{(3 x+2)^4}+9250 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-85094 - 225262*x - 64725*x^2 + 124875*x^3))/(2 + 3*x)^4 + 9
250*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1555848

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Maple [A]  time = 0.018, size = 66, normalized size = 0.6 \[ 648\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{4625\, \left ( 1-2\,x \right ) ^{7/2}}{889056}}+{\frac{11675\, \left ( 1-2\,x \right ) ^{5/2}}{1143072}}+{\frac{16027\, \left ( 1-2\,x \right ) ^{3/2}}{489888}}-{\frac{4625\,\sqrt{1-2\,x}}{69984}} \right ) }+{\frac{4625\,\sqrt{21}}{777924}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

648*(-4625/889056*(1-2*x)^(7/2)+11675/1143072*(1-2*x)^(5/2)+16027/489888*(1-2*x)
^(3/2)-4625/69984*(1-2*x)^(1/2))/(-4-6*x)^4+4625/777924*arctanh(1/7*21^(1/2)*(1-
2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49584, size = 149, normalized size = 1.38 \[ -\frac{4625}{1555848} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{124875 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 245175 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 785323 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1586375 \, \sqrt{-2 \, x + 1}}{37044 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4625/1555848*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/37044*(124875*(-2*x + 1)^(7/2) - 245175*(-2*x + 1)^(5/2) - 785323*(
-2*x + 1)^(3/2) + 1586375*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 26
46*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.220487, size = 140, normalized size = 1.3 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (124875 \, x^{3} - 64725 \, x^{2} - 225262 \, x - 85094\right )} \sqrt{-2 \, x + 1} + 4625 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1555848 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1555848*sqrt(21)*(sqrt(21)*(124875*x^3 - 64725*x^2 - 225262*x - 85094)*sqrt(-2
*x + 1) + 4625*(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((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218876, size = 135, normalized size = 1.25 \[ -\frac{4625}{1555848} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{124875 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 245175 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 785323 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1586375 \, \sqrt{-2 \, x + 1}}{592704 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4625/1555848*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 1/592704*(124875*(2*x - 1)^3*sqrt(-2*x + 1) + 245175*(2*x - 1
)^2*sqrt(-2*x + 1) + 785323*(-2*x + 1)^(3/2) - 1586375*sqrt(-2*x + 1))/(3*x + 2)
^4