3.1935 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac{(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac{8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (3 x+2)}+\frac{8051 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(7/2)/(315*(2 + 3*x)^5) + (347*(1 - 2*x)^(7/2))/(8820*(2 + 3*x)^4) -
(8051*(1 - 2*x)^(5/2))/(26460*(2 + 3*x)^3) + (8051*(1 - 2*x)^(3/2))/(31752*(2 +
3*x)^2) - (8051*Sqrt[1 - 2*x])/(31752*(2 + 3*x)) + (8051*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/(15876*Sqrt[21])

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Rubi [A]  time = 0.151381, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac{(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac{8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (3 x+2)}+\frac{8051 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-(1 - 2*x)^(7/2)/(315*(2 + 3*x)^5) + (347*(1 - 2*x)^(7/2))/(8820*(2 + 3*x)^4) -
(8051*(1 - 2*x)^(5/2))/(26460*(2 + 3*x)^3) + (8051*(1 - 2*x)^(3/2))/(31752*(2 +
3*x)^2) - (8051*Sqrt[1 - 2*x])/(31752*(2 + 3*x)) + (8051*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/(15876*Sqrt[21])

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Rubi in Sympy [A]  time = 15.7018, size = 112, normalized size = 0.88 \[ \frac{347 \left (- 2 x + 1\right )^{\frac{7}{2}}}{8820 \left (3 x + 2\right )^{4}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{315 \left (3 x + 2\right )^{5}} - \frac{8051 \left (- 2 x + 1\right )^{\frac{5}{2}}}{26460 \left (3 x + 2\right )^{3}} + \frac{8051 \left (- 2 x + 1\right )^{\frac{3}{2}}}{31752 \left (3 x + 2\right )^{2}} - \frac{8051 \sqrt{- 2 x + 1}}{31752 \left (3 x + 2\right )} + \frac{8051 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{333396} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**6,x)

[Out]

347*(-2*x + 1)**(7/2)/(8820*(3*x + 2)**4) - (-2*x + 1)**(7/2)/(315*(3*x + 2)**5)
- 8051*(-2*x + 1)**(5/2)/(26460*(3*x + 2)**3) + 8051*(-2*x + 1)**(3/2)/(31752*(
3*x + 2)**2) - 8051*sqrt(-2*x + 1)/(31752*(3*x + 2)) + 8051*sqrt(21)*atanh(sqrt(
21)*sqrt(-2*x + 1)/7)/333396

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Mathematica [A]  time = 0.12269, size = 68, normalized size = 0.53 \[ \frac{80510 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} \left (7323345 x^4+12406455 x^3+8277204 x^2+2919346 x+503276\right )}{(3 x+2)^5}}{3333960} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

((-21*Sqrt[1 - 2*x]*(503276 + 2919346*x + 8277204*x^2 + 12406455*x^3 + 7323345*x
^4))/(2 + 3*x)^5 + 80510*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3333960

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ -3888\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{54247\, \left ( 1-2\,x \right ) ^{9/2}}{2286144}}+{\frac{12269\, \left ( 1-2\,x \right ) ^{7/2}}{69984}}-{\frac{16102\, \left ( 1-2\,x \right ) ^{5/2}}{32805}}+{\frac{394499\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}-{\frac{394499\,\sqrt{1-2\,x}}{1259712}} \right ) }+{\frac{8051\,\sqrt{21}}{333396}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^6,x)

[Out]

-3888*(-54247/2286144*(1-2*x)^(9/2)+12269/69984*(1-2*x)^(7/2)-16102/32805*(1-2*x
)^(5/2)+394499/629856*(1-2*x)^(3/2)-394499/1259712*(1-2*x)^(1/2))/(-4-6*x)^5+805
1/333396*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48828, size = 173, normalized size = 1.35 \[ -\frac{8051}{666792} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7323345 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 54106290 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 151487616 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 193304510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 96652255 \, \sqrt{-2 \, x + 1}}{79380 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

-8051/666792*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/79380*(7323345*(-2*x + 1)^(9/2) - 54106290*(-2*x + 1)^(7/2) + 151487
616*(-2*x + 1)^(5/2) - 193304510*(-2*x + 1)^(3/2) + 96652255*sqrt(-2*x + 1))/(24
3*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030
*x - 19208)

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Fricas [A]  time = 0.212591, size = 161, normalized size = 1.26 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt{-2 \, x + 1} - 40255 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3333960 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

-1/3333960*sqrt(21)*(sqrt(21)*(7323345*x^4 + 12406455*x^3 + 8277204*x^2 + 291934
6*x + 503276)*sqrt(-2*x + 1) - 40255*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 2
40*x + 32)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 8
10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213774, size = 157, normalized size = 1.23 \[ -\frac{8051}{666792} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{7323345 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 54106290 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 151487616 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 193304510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 96652255 \, \sqrt{-2 \, x + 1}}{2540160 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="giac")

[Out]

-8051/666792*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 1/2540160*(7323345*(2*x - 1)^4*sqrt(-2*x + 1) + 54106290*(2*x
- 1)^3*sqrt(-2*x + 1) + 151487616*(2*x - 1)^2*sqrt(-2*x + 1) - 193304510*(-2*x +
1)^(3/2) + 96652255*sqrt(-2*x + 1))/(3*x + 2)^5