3.1863 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=128 \[ \frac{23 (1-2 x)^{5/2}}{588 (3 x+2)^4}-\frac{(1-2 x)^{5/2}}{315 (3 x+2)^5}-\frac{4693 (1-2 x)^{3/2}}{15876 (3 x+2)^3}-\frac{4693 \sqrt{1-2 x}}{222264 (3 x+2)}+\frac{4693 \sqrt{1-2 x}}{31752 (3 x+2)^2}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(5/2)/(315*(2 + 3*x)^5) + (23*(1 - 2*x)^(5/2))/(588*(2 + 3*x)^4) - (4
693*(1 - 2*x)^(3/2))/(15876*(2 + 3*x)^3) + (4693*Sqrt[1 - 2*x])/(31752*(2 + 3*x)
^2) - (4693*Sqrt[1 - 2*x])/(222264*(2 + 3*x)) - (4693*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(111132*Sqrt[21])

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Rubi [A]  time = 0.138231, antiderivative size = 128, 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 \[ \frac{23 (1-2 x)^{5/2}}{588 (3 x+2)^4}-\frac{(1-2 x)^{5/2}}{315 (3 x+2)^5}-\frac{4693 (1-2 x)^{3/2}}{15876 (3 x+2)^3}-\frac{4693 \sqrt{1-2 x}}{222264 (3 x+2)}+\frac{4693 \sqrt{1-2 x}}{31752 (3 x+2)^2}-\frac{4693 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)^(5/2)/(315*(2 + 3*x)^5) + (23*(1 - 2*x)^(5/2))/(588*(2 + 3*x)^4) - (4
693*(1 - 2*x)^(3/2))/(15876*(2 + 3*x)^3) + (4693*Sqrt[1 - 2*x])/(31752*(2 + 3*x)
^2) - (4693*Sqrt[1 - 2*x])/(222264*(2 + 3*x)) - (4693*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(111132*Sqrt[21])

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Rubi in Sympy [A]  time = 14.4003, size = 112, normalized size = 0.88 \[ \frac{23 \left (- 2 x + 1\right )^{\frac{5}{2}}}{588 \left (3 x + 2\right )^{4}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{315 \left (3 x + 2\right )^{5}} - \frac{4693 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15876 \left (3 x + 2\right )^{3}} - \frac{4693 \sqrt{- 2 x + 1}}{222264 \left (3 x + 2\right )} + \frac{4693 \sqrt{- 2 x + 1}}{31752 \left (3 x + 2\right )^{2}} - \frac{4693 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2333772} \]

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)**6,x)

[Out]

23*(-2*x + 1)**(5/2)/(588*(3*x + 2)**4) - (-2*x + 1)**(5/2)/(315*(3*x + 2)**5) -
4693*(-2*x + 1)**(3/2)/(15876*(3*x + 2)**3) - 4693*sqrt(-2*x + 1)/(222264*(3*x
+ 2)) + 4693*sqrt(-2*x + 1)/(31752*(3*x + 2)**2) - 4693*sqrt(21)*atanh(sqrt(21)*
sqrt(-2*x + 1)/7)/2333772

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Mathematica [A]  time = 0.122395, size = 68, normalized size = 0.53 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (1900665 x^4-5801265 x^3-8540988 x^2-2143262 x+292028\right )}{(3 x+2)^5}-46930 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{23337720} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(292028 - 2143262*x - 8540988*x^2 - 5801265*x^3 + 1900665*x^
4))/(2 + 3*x)^5 - 46930*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/23337720

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Maple [A]  time = 0.019, size = 75, normalized size = 0.6 \[ -3888\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{4693\, \left ( 1-2\,x \right ) ^{9/2}}{5334336}}-{\frac{907\, \left ( 1-2\,x \right ) ^{7/2}}{489888}}+{\frac{6119\, \left ( 1-2\,x \right ) ^{5/2}}{229635}}-{\frac{32851\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}+{\frac{32851\,\sqrt{1-2\,x}}{1259712}} \right ) }-{\frac{4693\,\sqrt{21}}{2333772}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3888*(-4693/5334336*(1-2*x)^(9/2)-907/489888*(1-2*x)^(7/2)+6119/229635*(1-2*x)^
(5/2)-32851/629856*(1-2*x)^(3/2)+32851/1259712*(1-2*x)^(1/2))/(-4-6*x)^5-4693/23
33772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.5132, size = 173, normalized size = 1.35 \[ \frac{4693}{4667544} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1900665 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 3999870 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 57567552 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 112678930 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 56339465 \, \sqrt{-2 \, x + 1}}{555660 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4693/4667544*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/555660*(1900665*(-2*x + 1)^(9/2) + 3999870*(-2*x + 1)^(7/2) - 575675
52*(-2*x + 1)^(5/2) + 112678930*(-2*x + 1)^(3/2) - 56339465*sqrt(-2*x + 1))/(243
*(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 = 0.222669, size = 161, normalized size = 1.26 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1900665 \, x^{4} - 5801265 \, x^{3} - 8540988 \, x^{2} - 2143262 \, x + 292028\right )} \sqrt{-2 \, x + 1} - 23465 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{23337720 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/23337720*sqrt(21)*(sqrt(21)*(1900665*x^4 - 5801265*x^3 - 8540988*x^2 - 214326
2*x + 292028)*sqrt(-2*x + 1) - 23465*(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)**(3/2)*(3+5*x)**2/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215296, size = 157, normalized size = 1.23 \[ \frac{4693}{4667544} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1900665 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 3999870 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 57567552 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 112678930 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 56339465 \, \sqrt{-2 \, x + 1}}{17781120 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4693/4667544*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 1/17781120*(1900665*(2*x - 1)^4*sqrt(-2*x + 1) - 3999870*(2*x
- 1)^3*sqrt(-2*x + 1) - 57567552*(2*x - 1)^2*sqrt(-2*x + 1) + 112678930*(-2*x +
1)^(3/2) - 56339465*sqrt(-2*x + 1))/(3*x + 2)^5