3.1952 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=108 \[ -\frac{27}{220} (1-2 x)^{11/2}+\frac{18}{25} (1-2 x)^{9/2}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{2 (1-2 x)^{5/2}}{3125}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{242 \sqrt{1-2 x}}{15625}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2))/3125
- (3897*(1 - 2*x)^(7/2))/3500 + (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))
/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15625

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Rubi [A]  time = 0.134463, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{27}{220} (1-2 x)^{11/2}+\frac{18}{25} (1-2 x)^{9/2}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{2 (1-2 x)^{5/2}}{3125}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{242 \sqrt{1-2 x}}{15625}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

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

[Out]

(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2))/3125
- (3897*(1 - 2*x)^(7/2))/3500 + (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))
/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15625

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Rubi in Sympy [A]  time = 12.4998, size = 95, normalized size = 0.88 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{11}{2}}}{220} + \frac{18 \left (- 2 x + 1\right )^{\frac{9}{2}}}{25} - \frac{3897 \left (- 2 x + 1\right )^{\frac{7}{2}}}{3500} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{3125} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9375} + \frac{242 \sqrt{- 2 x + 1}}{15625} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{78125} \]

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),x)

[Out]

-27*(-2*x + 1)**(11/2)/220 + 18*(-2*x + 1)**(9/2)/25 - 3897*(-2*x + 1)**(7/2)/35
00 + 2*(-2*x + 1)**(5/2)/3125 + 22*(-2*x + 1)**(3/2)/9375 + 242*sqrt(-2*x + 1)/1
5625 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/78125

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Mathematica [A]  time = 0.0938392, size = 66, normalized size = 0.61 \[ \frac{5 \sqrt{1-2 x} \left (14175000 x^5+6142500 x^4-15572250 x^3-3564885 x^2+7726195 x-1796318\right )-55902 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18046875} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-1796318 + 7726195*x - 3564885*x^2 - 15572250*x^3 + 6142500*x^
4 + 14175000*x^5) - 55902*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18046875

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Maple [A]  time = 0.008, size = 74, normalized size = 0.7 \[{\frac{22}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{3125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{3897}{3500} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{18}{25} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{27}{220} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{242\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{15625}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/9375*(1-2*x)^(3/2)+2/3125*(1-2*x)^(5/2)-3897/3500*(1-2*x)^(7/2)+18/25*(1-2*x)
^(9/2)-27/220*(1-2*x)^(11/2)-242/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)+242/15625*(1-2*x)^(1/2)

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Maxima [A]  time = 1.51159, size = 123, normalized size = 1.14 \[ -\frac{27}{220} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{18}{25} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{3897}{3500} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{15625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/220*(-2*x + 1)^(11/2) + 18/25*(-2*x + 1)^(9/2) - 3897/3500*(-2*x + 1)^(7/2)
+ 2/3125*(-2*x + 1)^(5/2) + 22/9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(-(
sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*
x + 1)

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Fricas [A]  time = 0.212069, size = 105, normalized size = 0.97 \[ \frac{1}{18046875} \, \sqrt{5}{\left (\sqrt{5}{\left (14175000 \, x^{5} + 6142500 \, x^{4} - 15572250 \, x^{3} - 3564885 \, x^{2} + 7726195 \, x - 1796318\right )} \sqrt{-2 \, x + 1} + 27951 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18046875*sqrt(5)*(sqrt(5)*(14175000*x^5 + 6142500*x^4 - 15572250*x^3 - 3564885
*x^2 + 7726195*x - 1796318)*sqrt(-2*x + 1) + 27951*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 = 18.03, size = 134, normalized size = 1.24 \[ - \frac{27 \left (- 2 x + 1\right )^{\frac{11}{2}}}{220} + \frac{18 \left (- 2 x + 1\right )^{\frac{9}{2}}}{25} - \frac{3897 \left (- 2 x + 1\right )^{\frac{7}{2}}}{3500} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{3125} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9375} + \frac{242 \sqrt{- 2 x + 1}}{15625} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*(-2*x + 1)**(11/2)/220 + 18*(-2*x + 1)**(9/2)/25 - 3897*(-2*x + 1)**(7/2)/35
00 + 2*(-2*x + 1)**(5/2)/3125 + 22*(-2*x + 1)**(3/2)/9375 + 242*sqrt(-2*x + 1)/1
5625 + 2662*Piecewise((-sqrt(55)*acoth(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))/1562
5

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GIAC/XCAS [A]  time = 0.213471, size = 165, normalized size = 1.53 \[ \frac{27}{220} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{18}{25} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{3897}{3500} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{3125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{78125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{15625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/220*(2*x - 1)^5*sqrt(-2*x + 1) + 18/25*(2*x - 1)^4*sqrt(-2*x + 1) + 3897/3500
*(2*x - 1)^3*sqrt(-2*x + 1) + 2/3125*(2*x - 1)^2*sqrt(-2*x + 1) + 22/9375*(-2*x
+ 1)^(3/2) + 121/78125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqr
t(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*x + 1)