3.1821 \(\int \frac{\sqrt{1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{5 (5 x+3)}+\frac{11}{75} \sqrt{1-2 x} (3 x+2)^4+\frac{64 \sqrt{1-2 x} (3 x+2)^3}{2625}-\frac{172 \sqrt{1-2 x} (3 x+2)^2}{3125}-\frac{4 \sqrt{1-2 x} (3625 x+10998)}{15625}-\frac{328 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]

[Out]

(-172*Sqrt[1 - 2*x]*(2 + 3*x)^2)/3125 + (64*Sqrt[1 - 2*x]*(2 + 3*x)^3)/2625 + (1
1*Sqrt[1 - 2*x]*(2 + 3*x)^4)/75 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(5*(3 + 5*x)) - (4
*Sqrt[1 - 2*x]*(10998 + 3625*x))/15625 - (328*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(15625*Sqrt[55])

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Rubi [A]  time = 0.260691, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{5 (5 x+3)}+\frac{11}{75} \sqrt{1-2 x} (3 x+2)^4+\frac{64 \sqrt{1-2 x} (3 x+2)^3}{2625}-\frac{172 \sqrt{1-2 x} (3 x+2)^2}{3125}-\frac{4 \sqrt{1-2 x} (3625 x+10998)}{15625}-\frac{328 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-172*Sqrt[1 - 2*x]*(2 + 3*x)^2)/3125 + (64*Sqrt[1 - 2*x]*(2 + 3*x)^3)/2625 + (1
1*Sqrt[1 - 2*x]*(2 + 3*x)^4)/75 - (Sqrt[1 - 2*x]*(2 + 3*x)^5)/(5*(3 + 5*x)) - (4
*Sqrt[1 - 2*x]*(10998 + 3625*x))/15625 - (328*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(15625*Sqrt[55])

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Rubi in Sympy [A]  time = 34.3225, size = 116, normalized size = 0.87 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}}{5 \left (5 x + 3\right )} + \frac{11 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{75} + \frac{64 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{2625} - \frac{172 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{3125} - \frac{\sqrt{- 2 x + 1} \left (13702500 x + 41572440\right )}{14765625} - \frac{328 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{859375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**5/(5*(5*x + 3)) + 11*sqrt(-2*x + 1)*(3*x + 2)**4/75 +
64*sqrt(-2*x + 1)*(3*x + 2)**3/2625 - 172*sqrt(-2*x + 1)*(3*x + 2)**2/3125 - sq
rt(-2*x + 1)*(13702500*x + 41572440)/14765625 - 328*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/859375

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Mathematica [A]  time = 0.117275, size = 73, normalized size = 0.55 \[ \frac{\frac{55 \sqrt{1-2 x} \left (1181250 x^5+3864375 x^4+4760100 x^3+2225760 x^2-1133340 x-862072\right )}{5 x+3}-2296 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6015625} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-862072 - 1133340*x + 2225760*x^2 + 4760100*x^3 + 3864375*x^
4 + 1181250*x^5))/(3 + 5*x) - 2296*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/6
015625

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Maple [A]  time = 0.016, size = 81, normalized size = 0.6 \[{\frac{27}{200} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{8829}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{107109}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{144681}{25000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{6}{3125}\sqrt{1-2\,x}}+{\frac{2}{78125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{328\,\sqrt{55}}{859375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/200*(1-2*x)^(9/2)-8829/7000*(1-2*x)^(7/2)+107109/25000*(1-2*x)^(5/2)-144681/2
5000*(1-2*x)^(3/2)+6/3125*(1-2*x)^(1/2)+2/78125*(1-2*x)^(1/2)/(-6/5-2*x)-328/859
375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49628, size = 132, normalized size = 0.99 \[ \frac{27}{200} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{8829}{7000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{107109}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{144681}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{164}{859375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/200*(-2*x + 1)^(9/2) - 8829/7000*(-2*x + 1)^(7/2) + 107109/25000*(-2*x + 1)^(
5/2) - 144681/25000*(-2*x + 1)^(3/2) + 164/859375*sqrt(55)*log(-(sqrt(55) - 5*sq
rt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 6/3125*sqrt(-2*x + 1) - 1/15625*s
qrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.211285, size = 113, normalized size = 0.85 \[ \frac{\sqrt{55}{\left (\sqrt{55}{\left (1181250 \, x^{5} + 3864375 \, x^{4} + 4760100 \, x^{3} + 2225760 \, x^{2} - 1133340 \, x - 862072\right )} \sqrt{-2 \, x + 1} + 1148 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{6015625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6015625*sqrt(55)*(sqrt(55)*(1181250*x^5 + 3864375*x^4 + 4760100*x^3 + 2225760*
x^2 - 1133340*x - 862072)*sqrt(-2*x + 1) + 1148*(5*x + 3)*log((sqrt(55)*(5*x - 8
) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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Sympy [A]  time = 84.1751, size = 223, normalized size = 1.68 \[ \frac{27 \left (- 2 x + 1\right )^{\frac{9}{2}}}{200} - \frac{8829 \left (- 2 x + 1\right )^{\frac{7}{2}}}{7000} + \frac{107109 \left (- 2 x + 1\right )^{\frac{5}{2}}}{25000} - \frac{144681 \left (- 2 x + 1\right )^{\frac{3}{2}}}{25000} + \frac{6 \sqrt{- 2 x + 1}}{3125} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{15625} + \frac{326 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{15625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*(-2*x + 1)**(9/2)/200 - 8829*(-2*x + 1)**(7/2)/7000 + 107109*(-2*x + 1)**(5/2
)/25000 - 144681*(-2*x + 1)**(3/2)/25000 + 6*sqrt(-2*x + 1)/3125 - 44*Piecewise(
(sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/
11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x +
1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5)))/15625 + 326*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))/15625

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GIAC/XCAS [A]  time = 0.219316, size = 165, normalized size = 1.24 \[ \frac{27}{200} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{8829}{7000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{107109}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{144681}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{164}{859375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{6}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/200*(2*x - 1)^4*sqrt(-2*x + 1) + 8829/7000*(2*x - 1)^3*sqrt(-2*x + 1) + 10710
9/25000*(2*x - 1)^2*sqrt(-2*x + 1) - 144681/25000*(-2*x + 1)^(3/2) + 164/859375*
sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 6/3125*sqrt(-2*x + 1) - 1/15625*sqrt(-2*x + 1)/(5*x + 3)