3.1879 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^5}{3+5 x} \, dx\)

Optimal. Leaf size=121 \[ -\frac{243 (1-2 x)^{13/2}}{1040}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{2 (1-2 x)^{3/2}}{46875}+\frac{22 \sqrt{1-2 x}}{78125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(22*Sqrt[1 - 2*x])/78125 + (2*(1 - 2*x)^(3/2))/46875 - (4774713*(1 - 2*x)^(5/2))
/250000 + (806121*(1 - 2*x)^(7/2))/35000 - (5673*(1 - 2*x)^(9/2))/500 + (5751*(1
- 2*x)^(11/2))/2200 - (243*(1 - 2*x)^(13/2))/1040 - (22*Sqrt[11/5]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi [A]  time = 0.116987, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{243 (1-2 x)^{13/2}}{1040}+\frac{5751 (1-2 x)^{11/2}}{2200}-\frac{5673}{500} (1-2 x)^{9/2}+\frac{806121 (1-2 x)^{7/2}}{35000}-\frac{4774713 (1-2 x)^{5/2}}{250000}+\frac{2 (1-2 x)^{3/2}}{46875}+\frac{22 \sqrt{1-2 x}}{78125}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[1 - 2*x])/78125 + (2*(1 - 2*x)^(3/2))/46875 - (4774713*(1 - 2*x)^(5/2))
/250000 + (806121*(1 - 2*x)^(7/2))/35000 - (5673*(1 - 2*x)^(9/2))/500 + (5751*(1
- 2*x)^(11/2))/2200 - (243*(1 - 2*x)^(13/2))/1040 - (22*Sqrt[11/5]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi in Sympy [A]  time = 13.1163, size = 107, normalized size = 0.88 \[ - \frac{243 \left (- 2 x + 1\right )^{\frac{13}{2}}}{1040} + \frac{5751 \left (- 2 x + 1\right )^{\frac{11}{2}}}{2200} - \frac{5673 \left (- 2 x + 1\right )^{\frac{9}{2}}}{500} + \frac{806121 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35000} - \frac{4774713 \left (- 2 x + 1\right )^{\frac{5}{2}}}{250000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{46875} + \frac{22 \sqrt{- 2 x + 1}}{78125} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{390625} \]

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),x)

[Out]

-243*(-2*x + 1)**(13/2)/1040 + 5751*(-2*x + 1)**(11/2)/2200 - 5673*(-2*x + 1)**(
9/2)/500 + 806121*(-2*x + 1)**(7/2)/35000 - 4774713*(-2*x + 1)**(5/2)/250000 + 2
*(-2*x + 1)**(3/2)/46875 + 22*sqrt(-2*x + 1)/78125 - 22*sqrt(55)*atanh(sqrt(55)*
sqrt(-2*x + 1)/11)/390625

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Mathematica [A]  time = 0.0989947, size = 71, normalized size = 0.59 \[ \frac{-5 \sqrt{1-2 x} \left (3508312500 x^6+9100350000 x^5+6683000625 x^4-1659418875 x^3-4276774170 x^2-1321809935 x+1180568944\right )-66066 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1173046875} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(1180568944 - 1321809935*x - 4276774170*x^2 - 1659418875*x^3 +
6683000625*x^4 + 9100350000*x^5 + 3508312500*x^6) - 66066*Sqrt[55]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/1173046875

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Maple [A]  time = 0.01, size = 83, normalized size = 0.7 \[{\frac{2}{46875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{4774713}{250000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{806121}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{5673}{500} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{5751}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{243}{1040} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}-{\frac{22\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{78125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/46875*(1-2*x)^(3/2)-4774713/250000*(1-2*x)^(5/2)+806121/35000*(1-2*x)^(7/2)-56
73/500*(1-2*x)^(9/2)+5751/2200*(1-2*x)^(11/2)-243/1040*(1-2*x)^(13/2)-22/390625*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/78125*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49415, size = 135, normalized size = 1.12 \[ -\frac{243}{1040} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{5751}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{5673}{500} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{806121}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{4774713}{250000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{78125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/1040*(-2*x + 1)^(13/2) + 5751/2200*(-2*x + 1)^(11/2) - 5673/500*(-2*x + 1)^
(9/2) + 806121/35000*(-2*x + 1)^(7/2) - 4774713/250000*(-2*x + 1)^(5/2) + 2/4687
5*(-2*x + 1)^(3/2) + 11/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) + 22/78125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.212348, size = 112, normalized size = 0.93 \[ -\frac{1}{1173046875} \, \sqrt{5}{\left (\sqrt{5}{\left (3508312500 \, x^{6} + 9100350000 \, x^{5} + 6683000625 \, x^{4} - 1659418875 \, x^{3} - 4276774170 \, x^{2} - 1321809935 \, x + 1180568944\right )} \sqrt{-2 \, x + 1} - 33033 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1173046875*sqrt(5)*(sqrt(5)*(3508312500*x^6 + 9100350000*x^5 + 6683000625*x^4
- 1659418875*x^3 - 4276774170*x^2 - 1321809935*x + 1180568944)*sqrt(-2*x + 1) -
33033*sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 22.3684, size = 146, normalized size = 1.21 \[ - \frac{243 \left (- 2 x + 1\right )^{\frac{13}{2}}}{1040} + \frac{5751 \left (- 2 x + 1\right )^{\frac{11}{2}}}{2200} - \frac{5673 \left (- 2 x + 1\right )^{\frac{9}{2}}}{500} + \frac{806121 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35000} - \frac{4774713 \left (- 2 x + 1\right )^{\frac{5}{2}}}{250000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{46875} + \frac{22 \sqrt{- 2 x + 1}}{78125} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243*(-2*x + 1)**(13/2)/1040 + 5751*(-2*x + 1)**(11/2)/2200 - 5673*(-2*x + 1)**(
9/2)/500 + 806121*(-2*x + 1)**(7/2)/35000 - 4774713*(-2*x + 1)**(5/2)/250000 + 2
*(-2*x + 1)**(3/2)/46875 + 22*sqrt(-2*x + 1)/78125 + 242*Piecewise((-sqrt(55)*ac
oth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*
sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/78125

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GIAC/XCAS [A]  time = 0.214825, size = 186, normalized size = 1.54 \[ -\frac{243}{1040} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{5751}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{5673}{500} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{806121}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{4774713}{250000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{390625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{78125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/1040*(2*x - 1)^6*sqrt(-2*x + 1) - 5751/2200*(2*x - 1)^5*sqrt(-2*x + 1) - 56
73/500*(2*x - 1)^4*sqrt(-2*x + 1) - 806121/35000*(2*x - 1)^3*sqrt(-2*x + 1) - 47
74713/250000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/46875*(-2*x + 1)^(3/2) + 11/390625*s
qrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 22/78125*sqrt(-2*x + 1)