3.2169 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)
Optimal. Leaf size=76 \[ \frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{10}{363 (1-2 x)^{3/2}}-\frac{50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
[Out]
10/(363*(1 - 2*x)^(3/2)) + 50/(1331*Sqrt[1 - 2*x]) - 1/(11*(1 - 2*x)^(3/2)*(3 +
5*x)) - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331
_______________________________________________________________________________________
Rubi [A] time = 0.0764427, antiderivative size = 76, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235
\[ \frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{10}{363 (1-2 x)^{3/2}}-\frac{50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*(3 + 5*x)^2),x]
[Out]
10/(363*(1 - 2*x)^(3/2)) + 50/(1331*Sqrt[1 - 2*x]) - 1/(11*(1 - 2*x)^(3/2)*(3 +
5*x)) - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.34245, size = 65, normalized size = 0.86 \[ - \frac{50 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{14641} + \frac{50}{1331 \sqrt{- 2 x + 1}} + \frac{10}{363 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{1}{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(3+5*x)**2,x)
[Out]
-50*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/14641 + 50/(1331*sqrt(-2*x + 1))
+ 10/(363*(-2*x + 1)**(3/2)) - 1/(11*(-2*x + 1)**(3/2)*(5*x + 3))
_______________________________________________________________________________________
Mathematica [A] time = 0.107769, size = 58, normalized size = 0.76 \[ \frac{\frac{11 \left (-1500 x^2+400 x+417\right )}{(1-2 x)^{3/2} (5 x+3)}-150 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{43923} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*(3 + 5*x)^2),x]
[Out]
((11*(417 + 400*x - 1500*x^2))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 150*Sqrt[55]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]])/43923
_______________________________________________________________________________________
Maple [A] time = 0.016, size = 54, normalized size = 0.7 \[{\frac{4}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{40}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{10}{1331}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{50\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(3+5*x)^2,x)
[Out]
4/363/(1-2*x)^(3/2)+40/1331/(1-2*x)^(1/2)+10/1331*(1-2*x)^(1/2)/(-6/5-2*x)-50/14
641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
_______________________________________________________________________________________
Maxima [A] time = 1.49686, size = 100, normalized size = 1.32 \[ \frac{25}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (375 \,{\left (2 \, x - 1\right )}^{2} + 1100 \, x - 792\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
25/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 2/3993*(375*(2*x - 1)^2 + 1100*x - 792)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1
)^(3/2))
_______________________________________________________________________________________
Fricas [A] time = 0.216574, size = 119, normalized size = 1.57 \[ \frac{\sqrt{11}{\left (75 \, \sqrt{5}{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (1500 \, x^{2} - 400 \, x - 417\right )}\right )}}{43923 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
1/43923*sqrt(11)*(75*sqrt(5)*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x
- 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(1500*x^2 - 400*x - 417)
)/((10*x^2 + x - 3)*sqrt(-2*x + 1))
_______________________________________________________________________________________
Sympy [A] time = 6.87046, size = 2286, normalized size = 30.08 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-2*x)**(5/2)/(3+5*x)**2,x)
[Out]
Piecewise((15000*sqrt(5)*I*(x + 3/5)**3*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(3993
00*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3
/5)) - 7500*sqrt(5)*(x + 3/5)**3*log(110)/(399300*sqrt(11)*(x + 3/5)**3 - 878460
*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 7500*sqrt(5)*(x + 3/5)**3*
log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sq
rt(11)*(x + 3/5)) - 15000*sqrt(5)*(x + 3/5)**3*log(2)/(399300*sqrt(11)*(x + 3/5)
**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*(
x + 3/5)**3*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)) + 15000*sqrt(5)*(x + 3/5)**3*log(22)/(399300*sqrt(
11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 1
500*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 87846
0*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 33000*sqrt(5)*I*(x + 3/5)
**2*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sq
rt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 33000*sqrt(5)*(x + 3/5)**2*lo
g(22)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt
(11)*(x + 3/5)) - 16500*sqrt(5)*(x + 3/5)**2*log(10)/(399300*sqrt(11)*(x + 3/5)*
*3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 33000*sqrt(5)*(
x + 3/5)**2*log(2)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)) + 16500*sqrt(5)*(x + 3/5)**2*log(11)/(399300*sqrt(1
1)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 16
500*sqrt(5)*(x + 3/5)**2*log(110)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11
)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 2200*sqrt(55)*I*(x + 3/5)*sqrt(10*
x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqr
t(11)*(x + 3/5)) + 18150*sqrt(5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/
(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(
x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*log(110)/(399300*sqrt(11)*(x + 3/5)**3 - 8784
60*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*l
og(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqr
t(11)*(x + 3/5)) - 18150*sqrt(5)*(x + 3/5)*log(2)/(399300*sqrt(11)*(x + 3/5)**3
- 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*(x +
3/5)*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 4831
53*sqrt(11)*(x + 3/5)) + 18150*sqrt(5)*(x + 3/5)*log(22)/(399300*sqrt(11)*(x + 3
/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 363*sqrt(55
)*I*sqrt(10*x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)), 10*Abs(x + 3/5)/11 > 1), (-1500*sqrt(55)*sqrt(-10*
x + 5)*(x + 3/5)**2/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)) + 2200*sqrt(55)*sqrt(-10*x + 5)*(x + 3/5)/(399300*
sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)
) - 363*sqrt(55)*sqrt(-10*x + 5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)
*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*(x + 3/5)**3*log(x + 3
/5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(1
1)*(x + 3/5)) - 15000*sqrt(5)*(x + 3/5)**3*log(sqrt(-10*x/11 + 5/11) + 1)/(39930
0*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/
5)) - 7500*sqrt(5)*(x + 3/5)**3*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*s
qrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*(x + 3/5)**3*lo
g(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt
(11)*(x + 3/5)) + 7500*sqrt(5)*I*pi*(x + 3/5)**3/(399300*sqrt(11)*(x + 3/5)**3 -
878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*(x +
3/5)**2*log(x + 3/5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**
2 + 483153*sqrt(11)*(x + 3/5)) + 33000*sqrt(5)*(x + 3/5)**2*log(sqrt(-10*x/11 +
5/11) + 1)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153
*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*(x + 3/5)**2*log(10)/(399300*sqrt(11)*(x +
3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 16500*sqrt
(5)*(x + 3/5)**2*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/
5)**2 + 483153*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*I*pi*(x + 3/5)**2/(399300*sqr
t(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) +
9075*sqrt(5)*(x + 3/5)*log(x + 3/5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt
(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 18150*sqrt(5)*(x + 3/5)*log(sqr
t(-10*x/11 + 5/11) + 1)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5
)**2 + 483153*sqrt(11)*(x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*log(11)/(399300*sqrt(
11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9
075*sqrt(5)*(x + 3/5)*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x
+ 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*I*pi*(x + 3/5)/(399300*sq
rt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)),
True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.214339, size = 104, normalized size = 1.37 \[ \frac{25}{14641} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4 \,{\left (60 \, x - 41\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{25 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^2*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]
25/14641*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 4/3993*(60*x - 41)/((2*x - 1)*sqrt(-2*x + 1)) - 25/1331*sqrt(-2*x
+ 1)/(5*x + 3)