3.1951 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=121 \[ \frac{81}{520} (1-2 x)^{13/2}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{2 (1-2 x)^{5/2}}{15625}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{242 \sqrt{1-2 x}}{78125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(242*Sqrt[1 - 2*x])/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2))/156
25 - (136419*(1 - 2*x)^(7/2))/35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2
*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/78125

_______________________________________________________________________________________

Rubi [A]  time = 0.136763, antiderivative size = 121, 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 \[ \frac{81}{520} (1-2 x)^{13/2}-\frac{2889 (1-2 x)^{11/2}}{2200}+\frac{3819 (1-2 x)^{9/2}}{1000}-\frac{136419 (1-2 x)^{7/2}}{35000}+\frac{2 (1-2 x)^{5/2}}{15625}+\frac{22 (1-2 x)^{3/2}}{46875}+\frac{242 \sqrt{1-2 x}}{78125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

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

[Out]

(242*Sqrt[1 - 2*x])/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2))/156
25 - (136419*(1 - 2*x)^(7/2))/35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2
*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/78125

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4454, size = 107, normalized size = 0.88 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{13}{2}}}{520} - \frac{2889 \left (- 2 x + 1\right )^{\frac{11}{2}}}{2200} + \frac{3819 \left (- 2 x + 1\right )^{\frac{9}{2}}}{1000} - \frac{136419 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35000} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{15625} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{46875} + \frac{242 \sqrt{- 2 x + 1}}{78125} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{390625} \]

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),x)

[Out]

81*(-2*x + 1)**(13/2)/520 - 2889*(-2*x + 1)**(11/2)/2200 + 3819*(-2*x + 1)**(9/2
)/1000 - 136419*(-2*x + 1)**(7/2)/35000 + 2*(-2*x + 1)**(5/2)/15625 + 22*(-2*x +
1)**(3/2)/46875 + 242*sqrt(-2*x + 1)/78125 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-
2*x + 1)/11)/390625

_______________________________________________________________________________________

Mathematica [A]  time = 0.129296, size = 71, normalized size = 0.59 \[ \frac{5 \sqrt{1-2 x} \left (2338875000 x^6+2842087500 x^5-1540428750 x^4-2556079875 x^3+399578370 x^2+960784285 x-289133384\right )-726726 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1173046875} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-289133384 + 960784285*x + 399578370*x^2 - 2556079875*x^3 - 15
40428750*x^4 + 2842087500*x^5 + 2338875000*x^6) - 726726*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/1173046875

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 83, normalized size = 0.7 \[{\frac{22}{46875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{136419}{35000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{3819}{1000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2889}{2200} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{81}{520} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}-{\frac{242\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{78125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/46875*(1-2*x)^(3/2)+2/15625*(1-2*x)^(5/2)-136419/35000*(1-2*x)^(7/2)+3819/100
0*(1-2*x)^(9/2)-2889/2200*(1-2*x)^(11/2)+81/520*(1-2*x)^(13/2)-242/390625*arctan
h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/78125*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.4896, size = 135, normalized size = 1.12 \[ \frac{81}{520} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{2889}{2200} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{3819}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{136419}{35000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{78125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/520*(-2*x + 1)^(13/2) - 2889/2200*(-2*x + 1)^(11/2) + 3819/1000*(-2*x + 1)^(9
/2) - 136419/35000*(-2*x + 1)^(7/2) + 2/15625*(-2*x + 1)^(5/2) + 22/46875*(-2*x
+ 1)^(3/2) + 121/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) +
5*sqrt(-2*x + 1))) + 242/78125*sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.213604, size = 112, normalized size = 0.93 \[ \frac{1}{1173046875} \, \sqrt{5}{\left (\sqrt{5}{\left (2338875000 \, x^{6} + 2842087500 \, x^{5} - 1540428750 \, x^{4} - 2556079875 \, x^{3} + 399578370 \, x^{2} + 960784285 \, x - 289133384\right )} \sqrt{-2 \, x + 1} + 363363 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1173046875*sqrt(5)*(sqrt(5)*(2338875000*x^6 + 2842087500*x^5 - 1540428750*x^4
- 2556079875*x^3 + 399578370*x^2 + 960784285*x - 289133384)*sqrt(-2*x + 1) + 363
363*sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)))

_______________________________________________________________________________________

Sympy [A]  time = 23.4905, size = 146, normalized size = 1.21 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{13}{2}}}{520} - \frac{2889 \left (- 2 x + 1\right )^{\frac{11}{2}}}{2200} + \frac{3819 \left (- 2 x + 1\right )^{\frac{9}{2}}}{1000} - \frac{136419 \left (- 2 x + 1\right )^{\frac{7}{2}}}{35000} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{15625} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{46875} + \frac{242 \sqrt{- 2 x + 1}}{78125} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{78125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*(-2*x + 1)**(13/2)/520 - 2889*(-2*x + 1)**(11/2)/2200 + 3819*(-2*x + 1)**(9/2
)/1000 - 136419*(-2*x + 1)**(7/2)/35000 + 2*(-2*x + 1)**(5/2)/15625 + 22*(-2*x +
1)**(3/2)/46875 + 242*sqrt(-2*x + 1)/78125 + 2662*Piecewise((-sqrt(55)*acoth(sq
rt(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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216213, size = 186, normalized size = 1.54 \[ \frac{81}{520} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{2889}{2200} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{3819}{1000} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{136419}{35000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{46875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{390625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{78125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/520*(2*x - 1)^6*sqrt(-2*x + 1) + 2889/2200*(2*x - 1)^5*sqrt(-2*x + 1) + 3819/
1000*(2*x - 1)^4*sqrt(-2*x + 1) + 136419/35000*(2*x - 1)^3*sqrt(-2*x + 1) + 2/15
625*(2*x - 1)^2*sqrt(-2*x + 1) + 22/46875*(-2*x + 1)^(3/2) + 121/390625*sqrt(55)
*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 24
2/78125*sqrt(-2*x + 1)