3.2152 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=106 \[ \frac{729 (1-2 x)^{7/2}}{1120}-\frac{43011 (1-2 x)^{5/2}}{4000}+\frac{169209 (1-2 x)^{3/2}}{2000}-\frac{5992353 \sqrt{1-2 x}}{10000}-\frac{2739541}{3872 \sqrt{1-2 x}}+\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

[Out]

117649/(1056*(1 - 2*x)^(3/2)) - 2739541/(3872*Sqrt[1 - 2*x]) - (5992353*Sqrt[1 -
2*x])/10000 + (169209*(1 - 2*x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (7
29*(1 - 2*x)^(7/2))/1120 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(75625*Sqrt[55]
)

_______________________________________________________________________________________

Rubi [A]  time = 0.194148, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{729 (1-2 x)^{7/2}}{1120}-\frac{43011 (1-2 x)^{5/2}}{4000}+\frac{169209 (1-2 x)^{3/2}}{2000}-\frac{5992353 \sqrt{1-2 x}}{10000}-\frac{2739541}{3872 \sqrt{1-2 x}}+\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)),x]

[Out]

117649/(1056*(1 - 2*x)^(3/2)) - 2739541/(3872*Sqrt[1 - 2*x]) - (5992353*Sqrt[1 -
2*x])/10000 + (169209*(1 - 2*x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (7
29*(1 - 2*x)^(7/2))/1120 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(75625*Sqrt[55]
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.9086, size = 95, normalized size = 0.9 \[ \frac{729 \left (- 2 x + 1\right )^{\frac{7}{2}}}{1120} - \frac{43011 \left (- 2 x + 1\right )^{\frac{5}{2}}}{4000} + \frac{169209 \left (- 2 x + 1\right )^{\frac{3}{2}}}{2000} - \frac{5992353 \sqrt{- 2 x + 1}}{10000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{4159375} - \frac{2739541}{3872 \sqrt{- 2 x + 1}} + \frac{117649}{1056 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**6/(1-2*x)**(5/2)/(3+5*x),x)

[Out]

729*(-2*x + 1)**(7/2)/1120 - 43011*(-2*x + 1)**(5/2)/4000 + 169209*(-2*x + 1)**(
3/2)/2000 - 5992353*sqrt(-2*x + 1)/10000 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x +
1)/11)/4159375 - 2739541/(3872*sqrt(-2*x + 1)) + 117649/(1056*(-2*x + 1)**(3/2)
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.188278, size = 66, normalized size = 0.62 \[ \frac{-\frac{55 \left (33078375 x^5+190531440 x^4+611141355 x^3+2562785082 x^2-5374023537 x+1780047848\right )}{(1-2 x)^{3/2}}-42 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{87346875} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)),x]

[Out]

((-55*(1780047848 - 5374023537*x + 2562785082*x^2 + 611141355*x^3 + 190531440*x^
4 + 33078375*x^5))/(1 - 2*x)^(3/2) - 42*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/87346875

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 74, normalized size = 0.7 \[{\frac{117649}{1056} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{169209}{2000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{43011}{4000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{729}{1120} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{4159375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{2739541}{3872}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{5992353}{10000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x),x)

[Out]

117649/1056/(1-2*x)^(3/2)+169209/2000*(1-2*x)^(3/2)-43011/4000*(1-2*x)^(5/2)+729
/1120*(1-2*x)^(7/2)-2/4159375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2739
541/3872/(1-2*x)^(1/2)-5992353/10000*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.5082, size = 117, normalized size = 1.1 \[ \frac{729}{1120} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{43011}{4000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{169209}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{4159375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5992353}{10000} \, \sqrt{-2 \, x + 1} + \frac{16807 \,{\left (489 \, x - 206\right )}}{5808 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

729/1120*(-2*x + 1)^(7/2) - 43011/4000*(-2*x + 1)^(5/2) + 169209/2000*(-2*x + 1)
^(3/2) + 1/4159375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 5992353/10000*sqrt(-2*x + 1) + 16807/5808*(489*x - 206)/(-2*x +
1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.221144, size = 123, normalized size = 1.16 \[ \frac{\sqrt{55}{\left (21 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (33078375 \, x^{5} + 190531440 \, x^{4} + 611141355 \, x^{3} + 2562785082 \, x^{2} - 5374023537 \, x + 1780047848\right )}\right )}}{87346875 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

1/87346875*sqrt(55)*(21*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sq
rt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(33078375*x^5 + 190531440*x^4 + 611141355*x^
3 + 2562785082*x^2 - 5374023537*x + 1780047848))/((2*x - 1)*sqrt(-2*x + 1))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{6}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**6/(1-2*x)**(5/2)/(3+5*x),x)

[Out]

Integral((3*x + 2)**6/((-2*x + 1)**(5/2)*(5*x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218038, size = 150, normalized size = 1.42 \[ -\frac{729}{1120} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{43011}{4000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{169209}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{4159375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5992353}{10000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (489 \, x - 206\right )}}{5808 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^6/((5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

-729/1120*(2*x - 1)^3*sqrt(-2*x + 1) - 43011/4000*(2*x - 1)^2*sqrt(-2*x + 1) + 1
69209/2000*(-2*x + 1)^(3/2) + 1/4159375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqr
t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 5992353/10000*sqrt(-2*x + 1) - 168
07/5808*(489*x - 206)/((2*x - 1)*sqrt(-2*x + 1))