3.1891 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=148 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac{38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac{4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac{2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac{324 \sqrt{1-2 x}}{78125}-\frac{324 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(324*Sqrt[1 - 2*x])/78125 - (4016*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/48125 + (38*(1 -
2*x)^(3/2)*(2 + 3*x)^3)/4125 + (39*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/275 - ((1 - 2*x)
^(3/2)*(2 + 3*x)^5)/(5*(3 + 5*x)) - (2*(1 - 2*x)^(3/2)*(298462 + 204777*x))/5156
25 - (324*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi [A]  time = 0.26344, antiderivative size = 148, 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 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{3/2} (3 x+2)^4+\frac{38 (1-2 x)^{3/2} (3 x+2)^3}{4125}-\frac{4016 (1-2 x)^{3/2} (3 x+2)^2}{48125}-\frac{2 (1-2 x)^{3/2} (204777 x+298462)}{515625}+\frac{324 \sqrt{1-2 x}}{78125}-\frac{324 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

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

[Out]

(324*Sqrt[1 - 2*x])/78125 - (4016*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/48125 + (38*(1 -
2*x)^(3/2)*(2 + 3*x)^3)/4125 + (39*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/275 - ((1 - 2*x)
^(3/2)*(2 + 3*x)^5)/(5*(3 + 5*x)) - (2*(1 - 2*x)^(3/2)*(298462 + 204777*x))/5156
25 - (324*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi in Sympy [A]  time = 35.5685, size = 128, normalized size = 0.86 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{5}}{5 \left (5 x + 3\right )} + \frac{39 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}}{275} + \frac{38 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{4125} - \frac{4016 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{48125} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (129009510 x + 188031060\right )}{162421875} + \frac{324 \sqrt{- 2 x + 1}}{78125} - \frac{324 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{390625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(3*x + 2)**5/(5*(5*x + 3)) + 39*(-2*x + 1)**(3/2)*(3*x + 2)**
4/275 + 38*(-2*x + 1)**(3/2)*(3*x + 2)**3/4125 - 4016*(-2*x + 1)**(3/2)*(3*x + 2
)**2/48125 - (-2*x + 1)**(3/2)*(129009510*x + 188031060)/162421875 + 324*sqrt(-2
*x + 1)/78125 - 324*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/390625

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Mathematica [A]  time = 0.137739, size = 78, normalized size = 0.53 \[ \frac{-\frac{5 \sqrt{1-2 x} \left (106312500 x^6+270112500 x^5+181738125 x^4-76760550 x^3-135193430 x^2-2532130 x+23061496\right )}{5 x+3}-24948 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{30078125} \]

Antiderivative was successfully verified.

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

[Out]

((-5*Sqrt[1 - 2*x]*(23061496 - 2532130*x - 135193430*x^2 - 76760550*x^3 + 181738
125*x^4 + 270112500*x^5 + 106312500*x^6))/(3 + 5*x) - 24948*Sqrt[55]*ArcTanh[Sqr
t[5/11]*Sqrt[1 - 2*x]])/30078125

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Maple [A]  time = 0.017, size = 90, normalized size = 0.6 \[{\frac{243}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{981}{1000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{107109}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{434043}{125000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{326}{78125}\sqrt{1-2\,x}}+{\frac{22}{390625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{324\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/2200*(1-2*x)^(11/2)-981/1000*(1-2*x)^(9/2)+107109/35000*(1-2*x)^(7/2)-434043
/125000*(1-2*x)^(5/2)+2/3125*(1-2*x)^(3/2)+326/78125*(1-2*x)^(1/2)+22/390625*(1-
2*x)^(1/2)/(-6/5-2*x)-324/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.54204, size = 144, normalized size = 0.97 \[ \frac{243}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{981}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{107109}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{434043}{125000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{162}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{326}{78125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/2200*(-2*x + 1)^(11/2) - 981/1000*(-2*x + 1)^(9/2) + 107109/35000*(-2*x + 1)
^(7/2) - 434043/125000*(-2*x + 1)^(5/2) + 2/3125*(-2*x + 1)^(3/2) + 162/390625*s
qrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 326/
78125*sqrt(-2*x + 1) - 11/78125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.212137, size = 130, normalized size = 0.88 \[ \frac{\sqrt{5}{\left (12474 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{5}{\left (106312500 \, x^{6} + 270112500 \, x^{5} + 181738125 \, x^{4} - 76760550 \, x^{3} - 135193430 \, x^{2} - 2532130 \, x + 23061496\right )} \sqrt{-2 \, x + 1}\right )}}{30078125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30078125*sqrt(5)*(12474*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)
*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(5)*(106312500*x^6 + 270112500*x^5 + 181738125
*x^4 - 76760550*x^3 - 135193430*x^2 - 2532130*x + 23061496)*sqrt(-2*x + 1))/(5*x
+ 3)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216762, size = 186, normalized size = 1.26 \[ -\frac{243}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{981}{1000} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{107109}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{434043}{125000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{162}{390625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{326}{78125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/2200*(2*x - 1)^5*sqrt(-2*x + 1) - 981/1000*(2*x - 1)^4*sqrt(-2*x + 1) - 107
109/35000*(2*x - 1)^3*sqrt(-2*x + 1) - 434043/125000*(2*x - 1)^2*sqrt(-2*x + 1)
+ 2/3125*(-2*x + 1)^(3/2) + 162/390625*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt
(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 326/78125*sqrt(-2*x + 1) - 11/78125
*sqrt(-2*x + 1)/(5*x + 3)