3.1975 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=135 \[ -\frac{47 (1-2 x)^{3/2} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{954}{875} (1-2 x)^{3/2} (3 x+2)^2+\frac{3 (1-2 x)^{3/2} (2403 x+1618)}{6250}+\frac{5559 \sqrt{1-2 x}}{15625}-\frac{5559 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(5559*Sqrt[1 - 2*x])/15625 + (954*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/875 - ((1 - 2*x)^
(5/2)*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (47*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(25*(3 +
5*x)) + (3*(1 - 2*x)^(3/2)*(1618 + 2403*x))/6250 - (5559*Sqrt[11/5]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/15625

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Rubi [A]  time = 0.231648, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{47 (1-2 x)^{3/2} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{5/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{954}{875} (1-2 x)^{3/2} (3 x+2)^2+\frac{3 (1-2 x)^{3/2} (2403 x+1618)}{6250}+\frac{5559 \sqrt{1-2 x}}{15625}-\frac{5559 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

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

[Out]

(5559*Sqrt[1 - 2*x])/15625 + (954*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/875 - ((1 - 2*x)^
(5/2)*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (47*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(25*(3 +
5*x)) + (3*(1 - 2*x)^(3/2)*(1618 + 2403*x))/6250 - (5559*Sqrt[11/5]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/15625

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Rubi in Sympy [A]  time = 21.5326, size = 109, normalized size = 0.81 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{10 \left (5 x + 3\right )^{2}} - \frac{47 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2}}{275 \left (5 x + 3\right )} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (229725 x + 143616\right )}{481250} + \frac{1853 \left (- 2 x + 1\right )^{\frac{3}{2}}}{34375} + \frac{5559 \sqrt{- 2 x + 1}}{15625} - \frac{5559 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(3*x + 2)**3/(10*(5*x + 3)**2) - 47*(-2*x + 1)**(5/2)*(3*x +
2)**2/(275*(5*x + 3)) + (-2*x + 1)**(5/2)*(229725*x + 143616)/481250 + 1853*(-2*
x + 1)**(3/2)/34375 + 5559*sqrt(-2*x + 1)/15625 - 5559*sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/78125

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Mathematica [A]  time = 0.130807, size = 73, normalized size = 0.54 \[ \frac{\frac{5 \sqrt{1-2 x} \left (1350000 x^5-27000 x^4-1506900 x^3+1651030 x^2+2637795 x+770444\right )}{(5 x+3)^2}-77826 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1093750} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(770444 + 2637795*x + 1651030*x^2 - 1506900*x^3 - 27000*x^4 +
1350000*x^5))/(3 + 5*x)^2 - 77826*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/10
93750

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Maple [A]  time = 0.017, size = 84, normalized size = 0.6 \[ -{\frac{27}{875} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{54}{3125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{186}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{46}{125}\sqrt{1-2\,x}}+{\frac{22}{125\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{189}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2101}{250}\sqrt{1-2\,x}} \right ) }-{\frac{5559\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/875*(1-2*x)^(7/2)+54/3125*(1-2*x)^(5/2)+186/3125*(1-2*x)^(3/2)+46/125*(1-2*x
)^(1/2)+22/125*(189/50*(1-2*x)^(3/2)-2101/250*(1-2*x)^(1/2))/(-6-10*x)^2-5559/78
125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52987, size = 149, normalized size = 1.1 \[ -\frac{27}{875} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{54}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{186}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5559}{156250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{46}{125} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (945 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2101 \, \sqrt{-2 \, x + 1}\right )}}{15625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/875*(-2*x + 1)^(7/2) + 54/3125*(-2*x + 1)^(5/2) + 186/3125*(-2*x + 1)^(3/2)
+ 5559/156250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) + 46/125*sqrt(-2*x + 1) + 11/15625*(945*(-2*x + 1)^(3/2) - 2101*sqrt(-2
*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.228321, size = 135, normalized size = 1. \[ \frac{\sqrt{5}{\left (38913 \, \sqrt{11}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (1350000 \, x^{5} - 27000 \, x^{4} - 1506900 \, x^{3} + 1651030 \, x^{2} + 2637795 \, x + 770444\right )} \sqrt{-2 \, x + 1}\right )}}{1093750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1093750*sqrt(5)*(38913*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)*(1350000*x^5 - 27000*x^4 - 150690
0*x^3 + 1651030*x^2 + 2637795*x + 770444)*sqrt(-2*x + 1))/(25*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)**3/(3+5*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217299, size = 159, normalized size = 1.18 \[ \frac{27}{875} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{54}{3125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{186}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5559}{156250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{46}{125} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (945 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2101 \, \sqrt{-2 \, x + 1}\right )}}{62500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/875*(2*x - 1)^3*sqrt(-2*x + 1) + 54/3125*(2*x - 1)^2*sqrt(-2*x + 1) + 186/312
5*(-2*x + 1)^(3/2) + 5559/156250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 46/125*sqrt(-2*x + 1) + 11/62500*(945*(-2
*x + 1)^(3/2) - 2101*sqrt(-2*x + 1))/(5*x + 3)^2