3.1953 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx\)

Optimal. Leaf size=95 \[ \frac{1}{10} (1-2 x)^{9/2}-\frac{111}{350} (1-2 x)^{7/2}+\frac{2}{625} (1-2 x)^{5/2}+\frac{22 (1-2 x)^{3/2}}{1875}+\frac{242 \sqrt{1-2 x}}{3125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(242*Sqrt[1 - 2*x])/3125 + (22*(1 - 2*x)^(3/2))/1875 + (2*(1 - 2*x)^(5/2))/625 -
(111*(1 - 2*x)^(7/2))/350 + (1 - 2*x)^(9/2)/10 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/3125

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Rubi [A]  time = 0.129542, antiderivative size = 95, 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 \[ \frac{1}{10} (1-2 x)^{9/2}-\frac{111}{350} (1-2 x)^{7/2}+\frac{2}{625} (1-2 x)^{5/2}+\frac{22 (1-2 x)^{3/2}}{1875}+\frac{242 \sqrt{1-2 x}}{3125}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(242*Sqrt[1 - 2*x])/3125 + (22*(1 - 2*x)^(3/2))/1875 + (2*(1 - 2*x)^(5/2))/625 -
(111*(1 - 2*x)^(7/2))/350 + (1 - 2*x)^(9/2)/10 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/3125

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Rubi in Sympy [A]  time = 11.5608, size = 82, normalized size = 0.86 \[ \frac{\left (- 2 x + 1\right )^{\frac{9}{2}}}{10} - \frac{111 \left (- 2 x + 1\right )^{\frac{7}{2}}}{350} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{625} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1875} + \frac{242 \sqrt{- 2 x + 1}}{3125} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(9/2)/10 - 111*(-2*x + 1)**(7/2)/350 + 2*(-2*x + 1)**(5/2)/625 + 22*
(-2*x + 1)**(3/2)/1875 + 242*sqrt(-2*x + 1)/3125 - 242*sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/15625

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Mathematica [A]  time = 0.0834599, size = 61, normalized size = 0.64 \[ \frac{5 \sqrt{1-2 x} \left (105000 x^4-43500 x^3-91410 x^2+69995 x-8188\right )-5082 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{328125} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - 2*x]*(-8188 + 69995*x - 91410*x^2 - 43500*x^3 + 105000*x^4) - 5082*S
qrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/328125

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Maple [A]  time = 0.01, size = 65, normalized size = 0.7 \[{\frac{22}{1875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{111}{350} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{10} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{242\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{3125}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22/1875*(1-2*x)^(3/2)+2/625*(1-2*x)^(5/2)-111/350*(1-2*x)^(7/2)+1/10*(1-2*x)^(9/
2)-242/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/3125*(1-2*x)^(1/2
)

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Maxima [A]  time = 1.50439, size = 111, normalized size = 1.17 \[ \frac{1}{10} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{111}{350} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(-2*x + 1)^(9/2) - 111/350*(-2*x + 1)^(7/2) + 2/625*(-2*x + 1)^(5/2) + 22/1
875*(-2*x + 1)^(3/2) + 121/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) + 242/3125*sqrt(-2*x + 1)

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Fricas [A]  time = 0.217669, size = 99, normalized size = 1.04 \[ \frac{1}{328125} \, \sqrt{5}{\left (\sqrt{5}{\left (105000 \, x^{4} - 43500 \, x^{3} - 91410 \, x^{2} + 69995 \, x - 8188\right )} \sqrt{-2 \, x + 1} + 2541 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/328125*sqrt(5)*(sqrt(5)*(105000*x^4 - 43500*x^3 - 91410*x^2 + 69995*x - 8188)*
sqrt(-2*x + 1) + 2541*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 = 13.6067, size = 121, normalized size = 1.27 \[ \frac{\left (- 2 x + 1\right )^{\frac{9}{2}}}{10} - \frac{111 \left (- 2 x + 1\right )^{\frac{7}{2}}}{350} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{625} + \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{1875} + \frac{242 \sqrt{- 2 x + 1}}{3125} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(9/2)/10 - 111*(-2*x + 1)**(7/2)/350 + 2*(-2*x + 1)**(5/2)/625 + 22*
(-2*x + 1)**(3/2)/1875 + 242*sqrt(-2*x + 1)/3125 + 2662*Piecewise((-sqrt(55)*aco
th(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/3125

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GIAC/XCAS [A]  time = 0.214017, size = 143, normalized size = 1.51 \[ \frac{1}{10} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{111}{350} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{15625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{242}{3125} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(2*x - 1)^4*sqrt(-2*x + 1) + 111/350*(2*x - 1)^3*sqrt(-2*x + 1) + 2/625*(2*
x - 1)^2*sqrt(-2*x + 1) + 22/1875*(-2*x + 1)^(3/2) + 121/15625*sqrt(55)*ln(1/2*a
bs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/3125*sq
rt(-2*x + 1)