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3.2158 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=56 \[ \frac{10}{121 \sqrt{1-2 x}}+\frac{2}{33 (1-2 x)^{3/2}}-\frac{10}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

2/(33*(1 - 2*x)^(3/2)) + 10/(121*Sqrt[1 - 2*x]) - (10*Sqrt[5/11]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0567042, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{10}{121 \sqrt{1-2 x}}+\frac{2}{33 (1-2 x)^{3/2}}-\frac{10}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2/(33*(1 - 2*x)^(3/2)) + 10/(121*Sqrt[1 - 2*x]) - (10*Sqrt[5/11]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]])/121

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Rubi in Sympy [A]  time = 5.80181, size = 48, normalized size = 0.86 \[ - \frac{10 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1331} + \frac{10}{121 \sqrt{- 2 x + 1}} + \frac{2}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1331 + 10/(121*sqrt(-2*x + 1)) +
2/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0674819, size = 52, normalized size = 0.93 \[ -\frac{2 \left (330 x+15 \sqrt{55} (1-2 x)^{3/2} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-286\right )}{3993 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-286 + 330*x + 15*Sqrt[55]*(1 - 2*x)^(3/2)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
]))/(3993*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.012, size = 38, normalized size = 0.7 \[{\frac{2}{33} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{10\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{10}{121}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(5/2)/(3+5*x),x)

[Out]

2/33/(1-2*x)^(3/2)-10/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+10/121/
(1-2*x)^(1/2)

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Maxima [A]  time = 1.50324, size = 69, normalized size = 1.23 \[ \frac{5}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (15 \, x - 13\right )}}{363 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 4/363*(15*x - 13)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.245354, size = 105, normalized size = 1.88 \[ \frac{\sqrt{11}{\left (15 \, \sqrt{5}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 4 \, \sqrt{11}{\left (15 \, x - 13\right )}\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3993*sqrt(11)*(15*sqrt(5)*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x - 8) + 1
1*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 4*sqrt(11)*(15*x - 13))/((2*x - 1)*sqrt(-
2*x + 1))

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Sympy [A]  time = 4.1449, size = 1836, normalized size = 32.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((3000*sqrt(5)*I*(x + 3/5)**2*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(36300
*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1500*sqrt(
5)*(x + 3/5)**2*log(110)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5)
 + 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)**2*log(11)/(36300*sqrt(11)*(x + 3/5)
**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3000*sqrt(5)*(x + 3/5)**2*log
(2)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) +
1500*sqrt(5)*(x + 3/5)**2*log(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*
(x + 3/5) + 43923*sqrt(11)) + 3000*sqrt(5)*(x + 3/5)**2*log(22)/(36300*sqrt(11)*
(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 300*sqrt(55)*I*(x +
3/5)*sqrt(10*x - 5)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43
923*sqrt(11)) - 6600*sqrt(5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(363
00*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 6600*sqr
t(5)*(x + 3/5)*log(22)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) +
 43923*sqrt(11)) - 3300*sqrt(5)*(x + 3/5)*log(10)/(36300*sqrt(11)*(x + 3/5)**2 -
 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 6600*sqrt(5)*(x + 3/5)*log(2)/(363
00*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 3300*sqr
t(5)*(x + 3/5)*log(11)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) +
 43923*sqrt(11)) + 3300*sqrt(5)*(x + 3/5)*log(110)/(36300*sqrt(11)*(x + 3/5)**2
- 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 440*sqrt(55)*I*sqrt(10*x - 5)/(36
300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 3630*sq
rt(5)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(36300*sqrt(11)*(x + 3/5)**2 - 79860*
sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1815*sqrt(5)*log(110)/(36300*sqrt(11)*(x
+ 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1815*sqrt(5)*log(11)/(3
6300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3630*s
qrt(5)*log(2)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sq
rt(11)) + 1815*sqrt(5)*log(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x
+ 3/5) + 43923*sqrt(11)) + 3630*sqrt(5)*log(22)/(36300*sqrt(11)*(x + 3/5)**2 - 7
9860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (-300*sqrt(5
5)*sqrt(-10*x + 5)*(x + 3/5)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x +
3/5) + 43923*sqrt(11)) + 440*sqrt(55)*sqrt(-10*x + 5)/(36300*sqrt(11)*(x + 3/5)*
*2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(
x + 3/5)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11
)) - 3000*sqrt(5)*(x + 3/5)**2*log(sqrt(-10*x/11 + 5/11) + 1)/(36300*sqrt(11)*(x
 + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)
**2*log(11)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt
(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*s
qrt(11)*(x + 3/5) + 43923*sqrt(11)) + 1500*sqrt(5)*I*pi*(x + 3/5)**2/(36300*sqrt
(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3300*sqrt(5)*(x
 + 3/5)*log(x + 3/5)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 4
3923*sqrt(11)) + 6600*sqrt(5)*(x + 3/5)*log(sqrt(-10*x/11 + 5/11) + 1)/(36300*sq
rt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3300*sqrt(5)*
(x + 3/5)*log(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 4392
3*sqrt(11)) + 3300*sqrt(5)*(x + 3/5)*log(11)/(36300*sqrt(11)*(x + 3/5)**2 - 7986
0*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3300*sqrt(5)*I*pi*(x + 3/5)/(36300*sqrt
(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 1815*sqrt(5)*lo
g(x + 3/5)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(
11)) - 3630*sqrt(5)*log(sqrt(-10*x/11 + 5/11) + 1)/(36300*sqrt(11)*(x + 3/5)**2
- 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1815*sqrt(5)*log(11)/(36300*sqrt(
11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 1815*sqrt(5)*log
(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) +
 1815*sqrt(5)*I*pi/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 439
23*sqrt(11)), True))

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GIAC/XCAS [A]  time = 0.214574, size = 82, normalized size = 1.46 \[ \frac{5}{1331} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (15 \, x - 13\right )}}{363 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/1331*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 4/363*(15*x - 13)/((2*x - 1)*sqrt(-2*x + 1))

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