3.1948 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=161 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{209 \sqrt{1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac{146971 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7938 \sqrt{21}} \]

[Out]

(-209*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(756*(2 + 3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^
3)/(15*(2 + 3*x)^5) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^4) + (11*Sq
rt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) - (11*Sqrt[1 - 2*x]*(3911 + 6475*x))/(1
5876*(2 + 3*x)) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7938*Sqrt[21])

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Rubi [A]  time = 0.306201, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac{209 \sqrt{1-2 x} (5 x+3)^2}{756 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} (6475 x+3911)}{15876 (3 x+2)}-\frac{146971 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7938 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-209*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(756*(2 + 3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^
3)/(15*(2 + 3*x)^5) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^4) + (11*Sq
rt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)^3) - (11*Sqrt[1 - 2*x]*(3911 + 6475*x))/(1
5876*(2 + 3*x)) - (146971*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7938*Sqrt[21])

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Rubi in Sympy [A]  time = 21.1453, size = 124, normalized size = 0.77 \[ - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (59994 x + 37728\right )}{666792 \left (3 x + 2\right )^{3}} - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{126 \left (3 x + 2\right )^{4}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{15 \left (3 x + 2\right )^{5}} + \frac{146971 \left (- 2 x + 1\right )^{\frac{3}{2}}}{111132 \left (3 x + 2\right )} + \frac{146971 \sqrt{- 2 x + 1}}{55566} - \frac{146971 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{166698} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*(-2*x + 1)**(5/2)*(59994*x + 37728)/(666792*(3*x + 2)**3) - 11*(-2*x + 1)**(
5/2)*(5*x + 3)**2/(126*(3*x + 2)**4) - (-2*x + 1)**(5/2)*(5*x + 3)**3/(15*(3*x +
2)**5) + 146971*(-2*x + 1)**(3/2)/(111132*(3*x + 2)) + 146971*sqrt(-2*x + 1)/55
566 - 146971*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/166698

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Mathematica [A]  time = 0.136621, size = 73, normalized size = 0.45 \[ \frac{\frac{63 \sqrt{1-2 x} \left (26460000 x^5+126578745 x^4+207486855 x^3+157178184 x^2+56745266 x+7933096\right )}{(3 x+2)^5}-4409130 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{5000940} \]

Antiderivative was successfully verified.

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

[Out]

((63*Sqrt[1 - 2*x]*(7933096 + 56745266*x + 157178184*x^2 + 207486855*x^3 + 12657
8745*x^4 + 26460000*x^5))/(2 + 3*x)^5 - 4409130*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/5000940

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Maple [A]  time = 0.02, size = 84, normalized size = 0.5 \[{\frac{1000}{729}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{284287}{784} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{226727}{72} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{1383554}{135} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{9599737}{648} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{31200211}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{146971\,\sqrt{21}}{166698}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1000/729*(1-2*x)^(1/2)+8/3*(-284287/784*(1-2*x)^(9/2)+226727/72*(1-2*x)^(7/2)-13
83554/135*(1-2*x)^(5/2)+9599737/648*(1-2*x)^(3/2)-31200211/3888*(1-2*x)^(1/2))/(
-4-6*x)^5-146971/166698*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50387, size = 185, normalized size = 1.15 \[ \frac{146971}{333396} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1000}{729} \, \sqrt{-2 \, x + 1} + \frac{345408705 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2999598210 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 9762357024 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 14111613390 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7644051695 \, \sqrt{-2 \, x + 1}}{357210 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

146971/333396*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 1000/729*sqrt(-2*x + 1) + 1/357210*(345408705*(-2*x + 1)^(9/2) - 2999
598210*(-2*x + 1)^(7/2) + 9762357024*(-2*x + 1)^(5/2) - 14111613390*(-2*x + 1)^(
3/2) + 7644051695*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 = 0.212324, size = 167, normalized size = 1.04 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (26460000 \, x^{5} + 126578745 \, x^{4} + 207486855 \, x^{3} + 157178184 \, x^{2} + 56745266 \, x + 7933096\right )} \sqrt{-2 \, x + 1} + 734855 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1666980 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1666980*sqrt(21)*(sqrt(21)*(26460000*x^5 + 126578745*x^4 + 207486855*x^3 + 157
178184*x^2 + 56745266*x + 7933096)*sqrt(-2*x + 1) + 734855*(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3
*x + 2)))/(243*x^5 + 810*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)**3/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.248554, size = 169, normalized size = 1.05 \[ \frac{146971}{333396} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1000}{729} \, \sqrt{-2 \, x + 1} + \frac{345408705 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 2999598210 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 9762357024 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 14111613390 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7644051695 \, \sqrt{-2 \, x + 1}}{11430720 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

146971/333396*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 1000/729*sqrt(-2*x + 1) + 1/11430720*(345408705*(2*x - 1)^4*s
qrt(-2*x + 1) + 2999598210*(2*x - 1)^3*sqrt(-2*x + 1) + 9762357024*(2*x - 1)^2*s
qrt(-2*x + 1) - 14111613390*(-2*x + 1)^(3/2) + 7644051695*sqrt(-2*x + 1))/(3*x +
2)^5