3.216 \(\int \frac{1}{(1-x)^{7/2} x^5} \, dx\)

Optimal. Leaf size=127 \[ -\frac{429 \sqrt{1-x}}{20 x^4}+\frac{286}{15 \sqrt{1-x} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{2}{5 (1-x)^{5/2} x^4}-\frac{1001 \sqrt{1-x}}{40 x^3}-\frac{1001 \sqrt{1-x}}{32 x^2}-\frac{3003 \sqrt{1-x}}{64 x}-\frac{3003}{64} \tanh ^{-1}\left (\sqrt{1-x}\right ) \]

[Out]

2/(5*(1 - x)^(5/2)*x^4) + 26/(15*(1 - x)^(3/2)*x^4) + 286/(15*Sqrt[1 - x]*x^4) -
(429*Sqrt[1 - x])/(20*x^4) - (1001*Sqrt[1 - x])/(40*x^3) - (1001*Sqrt[1 - x])/(
32*x^2) - (3003*Sqrt[1 - x])/(64*x) - (3003*ArcTanh[Sqrt[1 - x]])/64

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Rubi [A]  time = 0.101853, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{429 \sqrt{1-x}}{20 x^4}+\frac{286}{15 \sqrt{1-x} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{2}{5 (1-x)^{5/2} x^4}-\frac{1001 \sqrt{1-x}}{40 x^3}-\frac{1001 \sqrt{1-x}}{32 x^2}-\frac{3003 \sqrt{1-x}}{64 x}-\frac{3003}{64} \tanh ^{-1}\left (\sqrt{1-x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(7/2)*x^5),x]

[Out]

2/(5*(1 - x)^(5/2)*x^4) + 26/(15*(1 - x)^(3/2)*x^4) + 286/(15*Sqrt[1 - x]*x^4) -
(429*Sqrt[1 - x])/(20*x^4) - (1001*Sqrt[1 - x])/(40*x^3) - (1001*Sqrt[1 - x])/(
32*x^2) - (3003*Sqrt[1 - x])/(64*x) - (3003*ArcTanh[Sqrt[1 - x]])/64

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Rubi in Sympy [A]  time = 5.27613, size = 104, normalized size = 0.82 \[ - \frac{3003 \operatorname{atanh}{\left (\sqrt{- x + 1} \right )}}{64} - \frac{3003 \sqrt{- x + 1}}{64 x} - \frac{1001 \sqrt{- x + 1}}{32 x^{2}} - \frac{1001 \sqrt{- x + 1}}{40 x^{3}} - \frac{429 \sqrt{- x + 1}}{20 x^{4}} + \frac{286}{15 x^{4} \sqrt{- x + 1}} + \frac{26}{15 x^{4} \left (- x + 1\right )^{\frac{3}{2}}} + \frac{2}{5 x^{4} \left (- x + 1\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(7/2)/x**5,x)

[Out]

-3003*atanh(sqrt(-x + 1))/64 - 3003*sqrt(-x + 1)/(64*x) - 1001*sqrt(-x + 1)/(32*
x**2) - 1001*sqrt(-x + 1)/(40*x**3) - 429*sqrt(-x + 1)/(20*x**4) + 286/(15*x**4*
sqrt(-x + 1)) + 26/(15*x**4*(-x + 1)**(3/2)) + 2/(5*x**4*(-x + 1)**(5/2))

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Mathematica [A]  time = 0.0770529, size = 61, normalized size = 0.48 \[ -\frac{-45045 x^6+105105 x^5-69069 x^4+6435 x^3+1430 x^2+520 x+240}{960 (1-x)^{5/2} x^4}-\frac{3003}{64} \tanh ^{-1}\left (\sqrt{1-x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(7/2)*x^5),x]

[Out]

-(240 + 520*x + 1430*x^2 + 6435*x^3 - 69069*x^4 + 105105*x^5 - 45045*x^6)/(960*(
1 - x)^(5/2)*x^4) - (3003*ArcTanh[Sqrt[1 - x]])/64

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Maple [A]  time = 0.026, size = 157, normalized size = 1.2 \[{\frac{1}{64} \left ( 1+\sqrt{1-x} \right ) ^{-4}}+{\frac{17}{96} \left ( 1+\sqrt{1-x} \right ) ^{-3}}+{\frac{159}{128} \left ( 1+\sqrt{1-x} \right ) ^{-2}}+{\frac{1083}{128} \left ( 1+\sqrt{1-x} \right ) ^{-1}}-{\frac{3003}{128}\ln \left ( 1+\sqrt{1-x} \right ) }+{\frac{2}{5} \left ( 1-x \right ) ^{-{\frac{5}{2}}}}+{\frac{10}{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}}}+30\,{\frac{1}{\sqrt{1-x}}}-{\frac{1}{64} \left ( -1+\sqrt{1-x} \right ) ^{-4}}+{\frac{17}{96} \left ( -1+\sqrt{1-x} \right ) ^{-3}}-{\frac{159}{128} \left ( -1+\sqrt{1-x} \right ) ^{-2}}+{\frac{1083}{128} \left ( -1+\sqrt{1-x} \right ) ^{-1}}+{\frac{3003}{128}\ln \left ( -1+\sqrt{1-x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(7/2)/x^5,x)

[Out]

1/64/(1+(1-x)^(1/2))^4+17/96/(1+(1-x)^(1/2))^3+159/128/(1+(1-x)^(1/2))^2+1083/12
8/(1+(1-x)^(1/2))-3003/128*ln(1+(1-x)^(1/2))+2/5/(1-x)^(5/2)+10/3/(1-x)^(3/2)+30
/(1-x)^(1/2)-1/64/(-1+(1-x)^(1/2))^4+17/96/(-1+(1-x)^(1/2))^3-159/128/(-1+(1-x)^
(1/2))^2+1083/128/(-1+(1-x)^(1/2))+3003/128*ln(-1+(1-x)^(1/2))

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Maxima [A]  time = 1.35221, size = 150, normalized size = 1.18 \[ \frac{45045 \,{\left (x - 1\right )}^{6} + 165165 \,{\left (x - 1\right )}^{5} + 219219 \,{\left (x - 1\right )}^{4} + 119691 \,{\left (x - 1\right )}^{3} + 18304 \,{\left (x - 1\right )}^{2} - 1664 \, x + 2048}{960 \,{\left ({\left (-x + 1\right )}^{\frac{13}{2}} - 4 \,{\left (-x + 1\right )}^{\frac{11}{2}} + 6 \,{\left (-x + 1\right )}^{\frac{9}{2}} - 4 \,{\left (-x + 1\right )}^{\frac{7}{2}} +{\left (-x + 1\right )}^{\frac{5}{2}}\right )}} - \frac{3003}{128} \, \log \left (\sqrt{-x + 1} + 1\right ) + \frac{3003}{128} \, \log \left (\sqrt{-x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(-x + 1)^(7/2)),x, algorithm="maxima")

[Out]

1/960*(45045*(x - 1)^6 + 165165*(x - 1)^5 + 219219*(x - 1)^4 + 119691*(x - 1)^3
+ 18304*(x - 1)^2 - 1664*x + 2048)/((-x + 1)^(13/2) - 4*(-x + 1)^(11/2) + 6*(-x
+ 1)^(9/2) - 4*(-x + 1)^(7/2) + (-x + 1)^(5/2)) - 3003/128*log(sqrt(-x + 1) + 1)
+ 3003/128*log(sqrt(-x + 1) - 1)

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Fricas [A]  time = 0.206486, size = 155, normalized size = 1.22 \[ \frac{90090 \, x^{6} - 210210 \, x^{5} + 138138 \, x^{4} - 12870 \, x^{3} - 45045 \,{\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \sqrt{-x + 1} \log \left (\sqrt{-x + 1} + 1\right ) + 45045 \,{\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \sqrt{-x + 1} \log \left (\sqrt{-x + 1} - 1\right ) - 2860 \, x^{2} - 1040 \, x - 480}{1920 \,{\left (x^{6} - 2 \, x^{5} + x^{4}\right )} \sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(-x + 1)^(7/2)),x, algorithm="fricas")

[Out]

1/1920*(90090*x^6 - 210210*x^5 + 138138*x^4 - 12870*x^3 - 45045*(x^6 - 2*x^5 + x
^4)*sqrt(-x + 1)*log(sqrt(-x + 1) + 1) + 45045*(x^6 - 2*x^5 + x^4)*sqrt(-x + 1)*
log(sqrt(-x + 1) - 1) - 2860*x^2 - 1040*x - 480)/((x^6 - 2*x^5 + x^4)*sqrt(-x +
1))

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Sympy [A]  time = 25.9014, size = 971, normalized size = 7.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x)**(7/2)/x**5,x)

[Out]

Piecewise((45045*I*x**7*asin(1/sqrt(x))/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*
x**4) - 45045*I*x**6*sqrt(x - 1)/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) -
135135*I*x**6*asin(1/sqrt(x))/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 1
05105*I*x**5*sqrt(x - 1)/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 135135*
I*x**5*asin(1/sqrt(x))/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) - 69069*I*x
**4*sqrt(x - 1)/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) - 45045*I*x**4*asi
n(1/sqrt(x))/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 6435*I*x**3*sqrt(x
- 1)/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 1430*I*x**2*sqrt(x - 1)/(96
0*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 520*I*x*sqrt(x - 1)/(960*x**7 - 288
0*x**6 + 2880*x**5 - 960*x**4) + 240*I*sqrt(x - 1)/(960*x**7 - 2880*x**6 + 2880*
x**5 - 960*x**4), Abs(x) > 1), (45045*x**7*log(x)/(1920*x**7 - 5760*x**6 + 5760*
x**5 - 1920*x**4) - 90090*x**7*log(sqrt(-x + 1) + 1)/(1920*x**7 - 5760*x**6 + 57
60*x**5 - 1920*x**4) + 45045*I*pi*x**7/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920
*x**4) - 90090*x**6*sqrt(-x + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4)
- 135135*x**6*log(x)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 270270*x
**6*log(sqrt(-x + 1) + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) - 1351
35*I*pi*x**6/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 210210*x**5*sqrt(
-x + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 135135*x**5*log(x)/(19
20*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) - 270270*x**5*log(sqrt(-x + 1) + 1)
/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 135135*I*pi*x**5/(1920*x**7 -
5760*x**6 + 5760*x**5 - 1920*x**4) - 138138*x**4*sqrt(-x + 1)/(1920*x**7 - 5760
*x**6 + 5760*x**5 - 1920*x**4) - 45045*x**4*log(x)/(1920*x**7 - 5760*x**6 + 5760
*x**5 - 1920*x**4) + 90090*x**4*log(sqrt(-x + 1) + 1)/(1920*x**7 - 5760*x**6 + 5
760*x**5 - 1920*x**4) - 45045*I*pi*x**4/(1920*x**7 - 5760*x**6 + 5760*x**5 - 192
0*x**4) + 12870*x**3*sqrt(-x + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4
) + 2860*x**2*sqrt(-x + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 104
0*x*sqrt(-x + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 480*sqrt(-x +
1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4), True))

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GIAC/XCAS [A]  time = 0.201556, size = 140, normalized size = 1.1 \[ \frac{2 \,{\left (225 \,{\left (x - 1\right )}^{2} - 25 \, x + 28\right )}}{15 \,{\left (x - 1\right )}^{2} \sqrt{-x + 1}} - \frac{3249 \,{\left (x - 1\right )}^{3} \sqrt{-x + 1} + 10633 \,{\left (x - 1\right )}^{2} \sqrt{-x + 1} - 11767 \,{\left (-x + 1\right )}^{\frac{3}{2}} + 4431 \, \sqrt{-x + 1}}{192 \, x^{4}} - \frac{3003}{128} \,{\rm ln}\left (\sqrt{-x + 1} + 1\right ) + \frac{3003}{128} \,{\rm ln}\left ({\left | \sqrt{-x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(-x + 1)^(7/2)),x, algorithm="giac")

[Out]

2/15*(225*(x - 1)^2 - 25*x + 28)/((x - 1)^2*sqrt(-x + 1)) - 1/192*(3249*(x - 1)^
3*sqrt(-x + 1) + 10633*(x - 1)^2*sqrt(-x + 1) - 11767*(-x + 1)^(3/2) + 4431*sqrt
(-x + 1))/x^4 - 3003/128*ln(sqrt(-x + 1) + 1) + 3003/128*ln(abs(sqrt(-x + 1) - 1
))