3.1862 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ \frac{277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac{14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (3 x+2)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(5/2)/(252*(2 + 3*x)^4) + (277*(1 - 2*x)^(5/2))/(5292*(2 + 3*x)^3) -
(14423*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)^2) + (14423*Sqrt[1 - 2*x])/(31752*(2 +
3*x)) - (14423*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(15876*Sqrt[21])

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Rubi [A]  time = 0.118817, antiderivative size = 108, 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 \[ \frac{277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac{14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (3 x+2)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(5/2)/(252*(2 + 3*x)^4) + (277*(1 - 2*x)^(5/2))/(5292*(2 + 3*x)^3) -
(14423*(1 - 2*x)^(3/2))/(31752*(2 + 3*x)^2) + (14423*Sqrt[1 - 2*x])/(31752*(2 +
3*x)) - (14423*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(15876*Sqrt[21])

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Rubi in Sympy [A]  time = 12.5541, size = 94, normalized size = 0.87 \[ \frac{277 \left (- 2 x + 1\right )^{\frac{5}{2}}}{5292 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{252 \left (3 x + 2\right )^{4}} - \frac{14423 \left (- 2 x + 1\right )^{\frac{3}{2}}}{31752 \left (3 x + 2\right )^{2}} + \frac{14423 \sqrt{- 2 x + 1}}{31752 \left (3 x + 2\right )} - \frac{14423 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{333396} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

277*(-2*x + 1)**(5/2)/(5292*(3*x + 2)**3) - (-2*x + 1)**(5/2)/(252*(3*x + 2)**4)
- 14423*(-2*x + 1)**(3/2)/(31752*(3*x + 2)**2) + 14423*sqrt(-2*x + 1)/(31752*(3
*x + 2)) - 14423*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/333396

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Mathematica [A]  time = 0.119303, size = 63, normalized size = 0.58 \[ \frac{\frac{21 \sqrt{1-2 x} \left (668979 x^3+988035 x^2+453730 x+60890\right )}{(3 x+2)^4}-28846 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(60890 + 453730*x + 988035*x^2 + 668979*x^3))/(2 + 3*x)^4 - 2
8846*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/666792

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Maple [A]  time = 0.019, size = 66, normalized size = 0.6 \[ 648\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{2753\, \left ( 1-2\,x \right ) ^{7/2}}{42336}}+{\frac{189667\, \left ( 1-2\,x \right ) ^{5/2}}{489888}}-{\frac{158653\, \left ( 1-2\,x \right ) ^{3/2}}{209952}}+{\frac{100961\,\sqrt{1-2\,x}}{209952}} \right ) }-{\frac{14423\,\sqrt{21}}{333396}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

648*(-2753/42336*(1-2*x)^(7/2)+189667/489888*(1-2*x)^(5/2)-158653/209952*(1-2*x)
^(3/2)+100961/209952*(1-2*x)^(1/2))/(-4-6*x)^4-14423/333396*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50726, size = 149, normalized size = 1.38 \[ \frac{14423}{666792} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{668979 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 3983007 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 7773997 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4947089 \, \sqrt{-2 \, x + 1}}{15876 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14423/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/15876*(668979*(-2*x + 1)^(7/2) - 3983007*(-2*x + 1)^(5/2) + 7773997*
(-2*x + 1)^(3/2) - 4947089*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2
646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.223621, size = 140, normalized size = 1.3 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (668979 \, x^{3} + 988035 \, x^{2} + 453730 \, x + 60890\right )} \sqrt{-2 \, x + 1} + 14423 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{666792 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/666792*sqrt(21)*(sqrt(21)*(668979*x^3 + 988035*x^2 + 453730*x + 60890)*sqrt(-2
*x + 1) + 14423*(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)**(3/2)*(3+5*x)**2/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213011, size = 135, normalized size = 1.25 \[ \frac{14423}{666792} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{668979 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 3983007 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 7773997 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 4947089 \, \sqrt{-2 \, x + 1}}{254016 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14423/666792*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 1/254016*(668979*(2*x - 1)^3*sqrt(-2*x + 1) + 3983007*(2*x - 1
)^2*sqrt(-2*x + 1) - 7773997*(-2*x + 1)^(3/2) + 4947089*sqrt(-2*x + 1))/(3*x + 2
)^4