3.2159 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=85 \[ \frac{272}{5929 \sqrt{1-2 x}}+\frac{4}{231 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.206412, antiderivative size = 85, 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 \[ \frac{272}{5929 \sqrt{1-2 x}}+\frac{4}{231 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 21.229, size = 73, normalized size = 0.86 \[ \frac{18 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{50 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{272}{5929 \sqrt{- 2 x + 1}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 50*sqrt(55)*atanh(sqrt(55)*sq
rt(-2*x + 1)/11)/1331 + 272/(5929*sqrt(-2*x + 1)) + 4/(231*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.205592, size = 77, normalized size = 0.91 \[ -\frac{4 (408 x-281)}{17787 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4*(-281 + 408*x))/(17787*(1 - 2*x)^(3/2)) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sq
rt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.019, size = 56, normalized size = 0.7 \[{\frac{4}{231} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{18\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{50\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{272}{5929}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/1331*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)

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Maxima [A]  time = 1.50284, size = 117, normalized size = 1.38 \[ \frac{25}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (408 \, x - 281\right )}}{17787 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 9/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 4/17787*(408*x - 281)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.235983, size = 186, normalized size = 2.19 \[ \frac{\sqrt{11} \sqrt{7}{\left (3675 \, \sqrt{7} \sqrt{5}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 3267 \, \sqrt{11} \sqrt{3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 4 \, \sqrt{11} \sqrt{7}{\left (408 \, x - 281\right )}\right )}}{1369599 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1369599*sqrt(11)*sqrt(7)*(3675*sqrt(7)*sqrt(5)*(2*x - 1)*sqrt(-2*x + 1)*log((s
qrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 3267*sqrt(11)*sqrt(3
)*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3
*x + 2)) + 4*sqrt(11)*sqrt(7)*(408*x - 281))/((2*x - 1)*sqrt(-2*x + 1))

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Sympy [A]  time = 12.2338, size = 105, normalized size = 1.24 \[ - \frac{50 \sqrt{55} i \operatorname{atan}{\left (\frac{\sqrt{110} \sqrt{x - \frac{1}{2}}}{11} \right )}}{1331} + \frac{18 \sqrt{21} i \operatorname{atan}{\left (\frac{\sqrt{42} \sqrt{x - \frac{1}{2}}}{7} \right )}}{343} - \frac{136 \sqrt{2} i}{5929 \sqrt{x - \frac{1}{2}}} + \frac{\sqrt{2} i}{231 \left (x - \frac{1}{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} i}{20 \left (x - \frac{1}{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-50*sqrt(55)*I*atan(sqrt(110)*sqrt(x - 1/2)/11)/1331 + 18*sqrt(21)*I*atan(sqrt(4
2)*sqrt(x - 1/2)/7)/343 - 136*sqrt(2)*I/(5929*sqrt(x - 1/2)) + sqrt(2)*I/(231*(x
- 1/2)**(3/2)) + sqrt(2)*I/(20*(x - 1/2)**(5/2))

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GIAC/XCAS [A]  time = 0.217937, size = 135, normalized size = 1.59 \[ \frac{25}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (408 \, x - 281\right )}}{17787 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 9/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 4/17787*(408*x - 281)/((2*x - 1)*sqrt(-2*x + 1))