3.2020 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{81}{280} (1-2 x)^{7/2}-\frac{2889 (1-2 x)^{5/2}}{1000}+\frac{11457 (1-2 x)^{3/2}}{1000}-\frac{136419 \sqrt{1-2 x}}{5000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

[Out]

(-136419*Sqrt[1 - 2*x])/5000 + (11457*(1 - 2*x)^(3/2))/1000 - (2889*(1 - 2*x)^(5
/2))/1000 + (81*(1 - 2*x)^(7/2))/280 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(62
5*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.0940398, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{81}{280} (1-2 x)^{7/2}-\frac{2889 (1-2 x)^{5/2}}{1000}+\frac{11457 (1-2 x)^{3/2}}{1000}-\frac{136419 \sqrt{1-2 x}}{5000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-136419*Sqrt[1 - 2*x])/5000 + (11457*(1 - 2*x)^(3/2))/1000 - (2889*(1 - 2*x)^(5
/2))/1000 + (81*(1 - 2*x)^(7/2))/280 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(62
5*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.00939, size = 71, normalized size = 0.89 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{7}{2}}}{280} - \frac{2889 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1000} + \frac{11457 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1000} - \frac{136419 \sqrt{- 2 x + 1}}{5000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{34375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*(-2*x + 1)**(7/2)/280 - 2889*(-2*x + 1)**(5/2)/1000 + 11457*(-2*x + 1)**(3/2)
/1000 - 136419*sqrt(-2*x + 1)/5000 - 2*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11
)/34375

_______________________________________________________________________________________

Mathematica [A]  time = 0.0884446, size = 56, normalized size = 0.7 \[ -\frac{3 \sqrt{1-2 x} \left (3375 x^3+11790 x^2+19095 x+26872\right )}{4375}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*(26872 + 19095*x + 11790*x^2 + 3375*x^3))/4375 - (2*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/(625*Sqrt[55])

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 56, normalized size = 0.7 \[{\frac{11457}{1000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2889}{1000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{81}{280} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{34375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{136419}{5000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11457/1000*(1-2*x)^(3/2)-2889/1000*(1-2*x)^(5/2)+81/280*(1-2*x)^(7/2)-2/34375*ar
ctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-136419/5000*(1-2*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.55474, size = 99, normalized size = 1.24 \[ \frac{81}{280} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2889}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{11457}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{34375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{136419}{5000} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/280*(-2*x + 1)^(7/2) - 2889/1000*(-2*x + 1)^(5/2) + 11457/1000*(-2*x + 1)^(3/
2) + 1/34375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 136419/5000*sqrt(-2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.255179, size = 85, normalized size = 1.06 \[ -\frac{1}{240625} \, \sqrt{55}{\left (3 \, \sqrt{55}{\left (3375 \, x^{3} + 11790 \, x^{2} + 19095 \, x + 26872\right )} \sqrt{-2 \, x + 1} - 7 \, \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/240625*sqrt(55)*(3*sqrt(55)*(3375*x^3 + 11790*x^2 + 19095*x + 26872)*sqrt(-2*
x + 1) - 7*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))

_______________________________________________________________________________________

Sympy [A]  time = 6.19791, size = 114, normalized size = 1.42 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{7}{2}}}{280} - \frac{2889 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1000} + \frac{11457 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1000} - \frac{136419 \sqrt{- 2 x + 1}}{5000} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{- 2 x + 1}} \right )}}{55} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{5}{11} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*(-2*x + 1)**(7/2)/280 - 2889*(-2*x + 1)**(5/2)/1000 + 11457*(-2*x + 1)**(3/2)
/1000 - 136419*sqrt(-2*x + 1)/5000 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)/(5*sq
rt(-2*x + 1)))/55, 1/(-2*x + 1) > 5/11), (-sqrt(55)*atanh(sqrt(55)/(5*sqrt(-2*x
+ 1)))/55, 1/(-2*x + 1) < 5/11))/625

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.236888, size = 122, normalized size = 1.52 \[ -\frac{81}{280} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2889}{1000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{11457}{1000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{34375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{136419}{5000} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/280*(2*x - 1)^3*sqrt(-2*x + 1) - 2889/1000*(2*x - 1)^2*sqrt(-2*x + 1) + 1145
7/1000*(-2*x + 1)^(3/2) + 1/34375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 136419/5000*sqrt(-2*x + 1)