3.1977 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=96 \[ -\frac{(1-2 x)^{7/2}}{110 (5 x+3)^2}-\frac{63 (1-2 x)^{5/2}}{550 (5 x+3)}-\frac{21}{275} (1-2 x)^{3/2}-\frac{63}{125} \sqrt{1-2 x}+\frac{63}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-63*Sqrt[1 - 2*x])/125 - (21*(1 - 2*x)^(3/2))/275 - (1 - 2*x)^(7/2)/(110*(3 + 5
*x)^2) - (63*(1 - 2*x)^(5/2))/(550*(3 + 5*x)) + (63*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/125

_______________________________________________________________________________________

Rubi [A]  time = 0.0992715, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(1-2 x)^{7/2}}{110 (5 x+3)^2}-\frac{63 (1-2 x)^{5/2}}{550 (5 x+3)}-\frac{21}{275} (1-2 x)^{3/2}-\frac{63}{125} \sqrt{1-2 x}+\frac{63}{125} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-63*Sqrt[1 - 2*x])/125 - (21*(1 - 2*x)^(3/2))/275 - (1 - 2*x)^(7/2)/(110*(3 + 5
*x)^2) - (63*(1 - 2*x)^(5/2))/(550*(3 + 5*x)) + (63*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/125

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.84733, size = 80, normalized size = 0.83 \[ - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{110 \left (5 x + 3\right )^{2}} - \frac{63 \left (- 2 x + 1\right )^{\frac{5}{2}}}{550 \left (5 x + 3\right )} - \frac{21 \left (- 2 x + 1\right )^{\frac{3}{2}}}{275} - \frac{63 \sqrt{- 2 x + 1}}{125} + \frac{63 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(7/2)/(110*(5*x + 3)**2) - 63*(-2*x + 1)**(5/2)/(550*(5*x + 3)) - 2
1*(-2*x + 1)**(3/2)/275 - 63*sqrt(-2*x + 1)/125 + 63*sqrt(55)*atanh(sqrt(55)*sqr
t(-2*x + 1)/11)/625

_______________________________________________________________________________________

Mathematica [A]  time = 0.112232, size = 63, normalized size = 0.66 \[ \frac{\frac{5 \sqrt{1-2 x} \left (400 x^3-2280 x^2-3795 x-1394\right )}{(5 x+3)^2}+126 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1250} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(-1394 - 3795*x - 2280*x^2 + 400*x^3))/(3 + 5*x)^2 + 126*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1250

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 66, normalized size = 0.7 \[ -{\frac{4}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{256}{625}\sqrt{1-2\,x}}-{\frac{44}{25\, \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{57}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{649}{100}\sqrt{1-2\,x}} \right ) }+{\frac{63\,\sqrt{55}}{625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/125*(1-2*x)^(3/2)-256/625*(1-2*x)^(1/2)-44/25*(-57/20*(1-2*x)^(3/2)+649/100*(
1-2*x)^(1/2))/(-6-10*x)^2+63/625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49943, size = 124, normalized size = 1.29 \[ -\frac{4}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{63}{1250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{256}{625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (285 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 649 \, \sqrt{-2 \, x + 1}\right )}}{625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/125*(-2*x + 1)^(3/2) - 63/1250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(s
qrt(55) + 5*sqrt(-2*x + 1))) - 256/625*sqrt(-2*x + 1) + 11/625*(285*(-2*x + 1)^(
3/2) - 649*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

_______________________________________________________________________________________

Fricas [A]  time = 0.215696, size = 122, normalized size = 1.27 \[ \frac{\sqrt{5}{\left (63 \, \sqrt{11}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (400 \, x^{3} - 2280 \, x^{2} - 3795 \, x - 1394\right )} \sqrt{-2 \, x + 1}\right )}}{1250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1250*sqrt(5)*(63*sqrt(11)*(25*x^2 + 30*x + 9)*log((sqrt(5)*(5*x - 8) - 5*sqrt(
11)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(400*x^3 - 2280*x^2 - 3795*x - 1394)*sq
rt(-2*x + 1))/(25*x^2 + 30*x + 9)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213426, size = 116, normalized size = 1.21 \[ -\frac{4}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{63}{1250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{256}{625} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (285 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 649 \, \sqrt{-2 \, x + 1}\right )}}{2500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/125*(-2*x + 1)^(3/2) - 63/1250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 256/625*sqrt(-2*x + 1) + 11/2500*(285*(-
2*x + 1)^(3/2) - 649*sqrt(-2*x + 1))/(5*x + 3)^2