3.1974 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac{127 (1-2 x)^{3/2} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{1117}{750} (1-2 x)^{3/2} (3 x+2)^3+\frac{1903 (1-2 x)^{3/2} (3 x+2)^2}{4375}+\frac{(1-2 x)^{3/2} (24939 x+734)}{93750}+\frac{11763 \sqrt{1-2 x}}{78125}-\frac{11763 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(11763*Sqrt[1 - 2*x])/78125 + (1903*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/4375 + (1117*(1
- 2*x)^(3/2)*(2 + 3*x)^3)/750 - ((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(10*(3 + 5*x)^2)
- (127*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/(50*(3 + 5*x)) + ((1 - 2*x)^(3/2)*(734 + 249
39*x))/93750 - (11763*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi [A]  time = 0.286376, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{127 (1-2 x)^{3/2} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{1117}{750} (1-2 x)^{3/2} (3 x+2)^3+\frac{1903 (1-2 x)^{3/2} (3 x+2)^2}{4375}+\frac{(1-2 x)^{3/2} (24939 x+734)}{93750}+\frac{11763 \sqrt{1-2 x}}{78125}-\frac{11763 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

(11763*Sqrt[1 - 2*x])/78125 + (1903*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/4375 + (1117*(1
- 2*x)^(3/2)*(2 + 3*x)^3)/750 - ((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(10*(3 + 5*x)^2)
- (127*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/(50*(3 + 5*x)) + ((1 - 2*x)^(3/2)*(734 + 249
39*x))/93750 - (11763*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi in Sympy [A]  time = 29.1388, size = 128, normalized size = 0.83 \[ - \frac{\left (- 938925 x + 1159092\right ) \left (- 2 x + 1\right )^{\frac{5}{2}}}{21656250} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{4}}{10 \left (5 x + 3\right )^{2}} - \frac{127 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{550 \left (5 x + 3\right )} + \frac{262 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2}}{1375} + \frac{3921 \left (- 2 x + 1\right )^{\frac{3}{2}}}{171875} + \frac{11763 \sqrt{- 2 x + 1}}{78125} - \frac{11763 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{390625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x)**3,x)

[Out]

-(-938925*x + 1159092)*(-2*x + 1)**(5/2)/21656250 - (-2*x + 1)**(5/2)*(3*x + 2)*
*4/(10*(5*x + 3)**2) - 127*(-2*x + 1)**(5/2)*(3*x + 2)**3/(550*(5*x + 3)) + 262*
(-2*x + 1)**(5/2)*(3*x + 2)**2/1375 + 3921*(-2*x + 1)**(3/2)/171875 + 11763*sqrt
(-2*x + 1)/78125 - 11763*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/390625

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Mathematica [A]  time = 0.14941, size = 78, normalized size = 0.5 \[ \frac{\frac{5 \sqrt{1-2 x} \left (15750000 x^6+15075000 x^5-16051500 x^4-11139550 x^3+9372960 x^2+6891315 x+871208\right )}{(5 x+3)^2}-164682 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5468750} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^3,x]

[Out]

((5*Sqrt[1 - 2*x]*(871208 + 6891315*x + 9372960*x^2 - 11139550*x^3 - 16051500*x^
4 + 15075000*x^5 + 15750000*x^6))/(3 + 5*x)^2 - 164682*Sqrt[55]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/5468750

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Maple [A]  time = 0.017, size = 93, normalized size = 0.6 \[{\frac{9}{250} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{1107}{8750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{108}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{76}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2404}{15625}\sqrt{1-2\,x}}+{\frac{44}{625\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{51}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2827}{500}\sqrt{1-2\,x}} \right ) }-{\frac{11763\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^3,x)

[Out]

9/250*(1-2*x)^(9/2)-1107/8750*(1-2*x)^(7/2)+108/15625*(1-2*x)^(5/2)+76/3125*(1-2
*x)^(3/2)+2404/15625*(1-2*x)^(1/2)+44/625*(51/20*(1-2*x)^(3/2)-2827/500*(1-2*x)^
(1/2))/(-6-10*x)^2-11763/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.55026, size = 161, normalized size = 1.04 \[ \frac{9}{250} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{1107}{8750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{108}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{76}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11763}{781250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2404}{15625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (1275 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2827 \, \sqrt{-2 \, x + 1}\right )}}{78125 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^3,x, algorithm="maxima")

[Out]

9/250*(-2*x + 1)^(9/2) - 1107/8750*(-2*x + 1)^(7/2) + 108/15625*(-2*x + 1)^(5/2)
+ 76/3125*(-2*x + 1)^(3/2) + 11763/781250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2404/15625*sqrt(-2*x + 1) + 11/78125*(12
75*(-2*x + 1)^(3/2) - 2827*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.215177, size = 142, normalized size = 0.92 \[ \frac{\sqrt{5}{\left (82341 \, \sqrt{11}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (15750000 \, x^{6} + 15075000 \, x^{5} - 16051500 \, x^{4} - 11139550 \, x^{3} + 9372960 \, x^{2} + 6891315 \, x + 871208\right )} \sqrt{-2 \, x + 1}\right )}}{5468750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^3,x, algorithm="fricas")

[Out]

1/5468750*sqrt(5)*(82341*sqrt(11)*(25*x^2 + 30*x + 9)*log((sqrt(5)*(5*x - 8) + 5
*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(15750000*x^6 + 15075000*x^5 - 16
051500*x^4 - 11139550*x^3 + 9372960*x^2 + 6891315*x + 871208)*sqrt(-2*x + 1))/(2
5*x^2 + 30*x + 9)

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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)*(2+3*x)**4/(3+5*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216581, size = 181, normalized size = 1.17 \[ \frac{9}{250} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{1107}{8750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{108}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{76}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11763}{781250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2404}{15625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (1275 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2827 \, \sqrt{-2 \, x + 1}\right )}}{312500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^3,x, algorithm="giac")

[Out]

9/250*(2*x - 1)^4*sqrt(-2*x + 1) + 1107/8750*(2*x - 1)^3*sqrt(-2*x + 1) + 108/15
625*(2*x - 1)^2*sqrt(-2*x + 1) + 76/3125*(-2*x + 1)^(3/2) + 11763/781250*sqrt(55
)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2
404/15625*sqrt(-2*x + 1) + 11/312500*(1275*(-2*x + 1)^(3/2) - 2827*sqrt(-2*x + 1
))/(5*x + 3)^2