3.2179 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=110 \[ -\frac{875 \sqrt{1-2 x}}{29282 (5 x+3)}-\frac{875 \sqrt{1-2 x}}{7986 (5 x+3)^2}+\frac{70}{363 \sqrt{1-2 x} (5 x+3)^2}+\frac{2}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{175 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 70/(363*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (875*S
qrt[1 - 2*x])/(7986*(3 + 5*x)^2) - (875*Sqrt[1 - 2*x])/(29282*(3 + 5*x)) - (175*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.102783, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{875 \sqrt{1-2 x}}{29282 (5 x+3)}-\frac{875 \sqrt{1-2 x}}{7986 (5 x+3)^2}+\frac{70}{363 \sqrt{1-2 x} (5 x+3)^2}+\frac{2}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac{175 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

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

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 70/(363*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (875*S
qrt[1 - 2*x])/(7986*(3 + 5*x)^2) - (875*Sqrt[1 - 2*x])/(29282*(3 + 5*x)) - (175*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi in Sympy [A]  time = 9.529, size = 95, normalized size = 0.86 \[ - \frac{875 \sqrt{- 2 x + 1}}{29282 \left (5 x + 3\right )} - \frac{875 \sqrt{- 2 x + 1}}{7986 \left (5 x + 3\right )^{2}} - \frac{175 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{161051} + \frac{70}{363 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} + \frac{2}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-875*sqrt(-2*x + 1)/(29282*(5*x + 3)) - 875*sqrt(-2*x + 1)/(7986*(5*x + 3)**2) -
175*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/161051 + 70/(363*sqrt(-2*x + 1)*
(5*x + 3)**2) + 2/(33*(-2*x + 1)**(3/2)*(5*x + 3)**2)

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Mathematica [A]  time = 0.104046, size = 66, normalized size = 0.6 \[ \frac{\frac{11 \sqrt{1-2 x} \left (-52500 x^3-17500 x^2+22995 x+4764\right )}{\left (10 x^2+x-3\right )^2}-1050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{966306} \]

Antiderivative was successfully verified.

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

[Out]

((11*Sqrt[1 - 2*x]*(4764 + 22995*x - 17500*x^2 - 52500*x^3))/(-3 + x + 10*x^2)^2
- 1050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/966306

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Maple [A]  time = 0.019, size = 66, normalized size = 0.6 \[{\frac{8}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{120}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{5000}{14641\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{11}{40} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{143}{200}\sqrt{1-2\,x}} \right ) }-{\frac{175\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/3993/(1-2*x)^(3/2)+120/14641/(1-2*x)^(1/2)+5000/14641*(11/40*(1-2*x)^(3/2)-143
/200*(1-2*x)^(1/2))/(-6-10*x)^2-175/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*
55^(1/2)

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Maxima [A]  time = 1.48717, size = 124, normalized size = 1.13 \[ \frac{175}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{13125 \,{\left (2 \, x - 1\right )}^{3} + 48125 \,{\left (2 \, x - 1\right )}^{2} + 67760 \, x - 44528}{43923 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

175/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 1/43923*(13125*(2*x - 1)^3 + 48125*(2*x - 1)^2 + 67760*x - 44528)/(25*(-
2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.218779, size = 144, normalized size = 1.31 \[ \frac{\sqrt{11}{\left (525 \, \sqrt{5}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{11}{\left (52500 \, x^{3} + 17500 \, x^{2} - 22995 \, x - 4764\right )}\right )}}{966306 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/966306*sqrt(11)*(525*sqrt(5)*(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1)*log((
sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(11)*(52500*x^3
+ 17500*x^2 - 22995*x - 4764))/((50*x^3 + 35*x^2 - 12*x - 9)*sqrt(-2*x + 1))

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Sympy [A]  time = 10.1048, size = 984, normalized size = 8.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-105000*sqrt(55)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*acos
h(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)
**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 52500*s
qrt(55)*I*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)/(96630600*sqrt(-1 +
11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*
(x + 3/5)**(153/2)) + 115500*sqrt(55)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(1
53/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))
*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2))
- 57750*sqrt(55)*I*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)/(96630600
*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(153/2)) + 577500*sqrt(2)*(x + 3/5)**77/(96630600*sqrt(-1 +
11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*
(x + 3/5)**(153/2)) - 847000*sqrt(2)*(x + 3/5)**76/(96630600*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)*
*(153/2)) + 139755*sqrt(2)*(x + 3/5)**75/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*
(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2))
+ 43923*sqrt(2)*(x + 3/5)**74/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**
(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)), 11*Abs(1/(
x + 3/5))/10 > 1), (105000*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(15
5/2)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x
+ 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 1
15500*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*asin(sqrt(110)/(
10*sqrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 10
6293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 577500*sqrt(2)*I*(x +
3/5)**77/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sq
rt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 847000*sqrt(2)*I*(x + 3/5)**76/(
96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/
(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 139755*sqrt(2)*I*(x + 3/5)**75/(96630600*s
qrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3
/5)))*(x + 3/5)**(153/2)) - 43923*sqrt(2)*I*(x + 3/5)**74/(96630600*sqrt(1 - 11/
(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x +
3/5)**(153/2)), True))

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GIAC/XCAS [A]  time = 0.23622, size = 120, normalized size = 1.09 \[ \frac{175}{322102} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (45 \, x - 28\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{25 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

175/322102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 16/43923*(45*x - 28)/((2*x - 1)*sqrt(-2*x + 1)) + 25/5324*(5*(-
2*x + 1)^(3/2) - 13*sqrt(-2*x + 1))/(5*x + 3)^2