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

Optimal. Leaf size=118 \[ -\frac{143}{96 (1-x)^{5/2} x^2}-\frac{13}{24 (1-x)^{5/2} x^3}-\frac{1}{4 (1-x)^{5/2} x^4}+\frac{3003}{64 \sqrt{1-x}}-\frac{429}{64 (1-x)^{5/2} x}+\frac{1001}{64 (1-x)^{3/2}}+\frac{3003}{320 (1-x)^{5/2}}-\frac{3003}{64} \tanh ^{-1}\left (\sqrt{1-x}\right ) \]

[Out]

3003/(320*(1 - x)^(5/2)) + 1001/(64*(1 - x)^(3/2)) + 3003/(64*Sqrt[1 - x]) - 1/(4*(1 - x)^(5/2)*x^4) - 13/(24*
(1 - x)^(5/2)*x^3) - 143/(96*(1 - x)^(5/2)*x^2) - 429/(64*(1 - x)^(5/2)*x) - (3003*ArcTanh[Sqrt[1 - x]])/64

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Rubi [A]  time = 0.0375306, antiderivative size = 127, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 206} \[ -\frac{1001 \sqrt{1-x}}{32 x^2}-\frac{1001 \sqrt{1-x}}{40 x^3}-\frac{429 \sqrt{1-x}}{20 x^4}+\frac{286}{15 \sqrt{1-x} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{2}{5 (1-x)^{5/2} x^4}-\frac{3003 \sqrt{1-x}}{64 x}-\frac{3003}{64} \tanh ^{-1}\left (\sqrt{1-x}\right ) \]

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-x)^{7/2} x^5} \, dx &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{13}{5} \int \frac{1}{(1-x)^{5/2} x^5} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{143}{15} \int \frac{1}{(1-x)^{3/2} x^5} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}+\frac{429}{5} \int \frac{1}{\sqrt{1-x} x^5} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 x^4}+\frac{3003}{40} \int \frac{1}{\sqrt{1-x} x^4} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 x^4}-\frac{1001 \sqrt{1-x}}{40 x^3}+\frac{1001}{16} \int \frac{1}{\sqrt{1-x} x^3} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 x^4}-\frac{1001 \sqrt{1-x}}{40 x^3}-\frac{1001 \sqrt{1-x}}{32 x^2}+\frac{3003}{64} \int \frac{1}{\sqrt{1-x} x^2} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 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}{128} \int \frac{1}{\sqrt{1-x} x} \, dx\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 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} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x}\right )\\ &=\frac{2}{5 (1-x)^{5/2} x^4}+\frac{26}{15 (1-x)^{3/2} x^4}+\frac{286}{15 \sqrt{1-x} x^4}-\frac{429 \sqrt{1-x}}{20 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 )\\ \end{align*}

Mathematica [C]  time = 0.0060539, size = 26, normalized size = 0.22 \[ \frac{2 \, _2F_1\left (-\frac{5}{2},5;-\frac{3}{2};1-x\right )}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(7/2)*x^5),x]

[Out]

(2*Hypergeometric2F1[-5/2, 5, -3/2, 1 - x])/(5*(1 - x)^(5/2))

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Maple [A]  time = 0.015, size = 157, normalized size = 1.3 \begin{align*}{\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 ) }-{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(7/2)/x^5,x)

[Out]

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))-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 = 0.936513, size = 150, normalized size = 1.27 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/x^5,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 + 2
048)/((-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*l
og(sqrt(-x + 1) + 1) + 3003/128*log(sqrt(-x + 1) - 1)

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Fricas [A]  time = 1.80789, size = 328, normalized size = 2.78 \begin{align*} -\frac{45045 \,{\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} \log \left (\sqrt{-x + 1} + 1\right ) - 45045 \,{\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} \log \left (\sqrt{-x + 1} - 1\right ) + 2 \,{\left (45045 \, x^{6} - 105105 \, x^{5} + 69069 \, x^{4} - 6435 \, x^{3} - 1430 \, x^{2} - 520 \, x - 240\right )} \sqrt{-x + 1}}{1920 \,{\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/x^5,x, algorithm="fricas")

[Out]

-1/1920*(45045*(x^7 - 3*x^6 + 3*x^5 - x^4)*log(sqrt(-x + 1) + 1) - 45045*(x^7 - 3*x^6 + 3*x^5 - x^4)*log(sqrt(
-x + 1) - 1) + 2*(45045*x^6 - 105105*x^5 + 69069*x^4 - 6435*x^3 - 1430*x^2 - 520*x - 240)*sqrt(-x + 1))/(x^7 -
3*x^6 + 3*x^5 - x^4)

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Sympy [C]  time = 15.7468, size = 971, normalized size = 8.23 \begin{align*} \text{result too large to display} \end{align*}

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) + 105105*I*x**5*sqrt(x - 1)/(960*x**7 - 2880*x**6 + 2880*x**5 - 960*x**4) + 135135*I*x**5*asi
n(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*asin(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)/(960*x**7 - 2880*x**6 +
2880*x**5 - 960*x**4) + 520*I*x*sqrt(x - 1)/(960*x**7 - 2880*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(1 - x) + 1)/(1920*x**7 - 5760*x**6 + 5760*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(1 - x)/(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
(1 - x) + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) - 135135*I*pi*x**6/(1920*x**7 - 5760*x**6 + 5760*
x**5 - 1920*x**4) + 210210*x**5*sqrt(1 - x)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 135135*x**5*log(
x)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) - 270270*x**5*log(sqrt(1 - x) + 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(
1 - x)/(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(1 - x) + 1)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) - 45045*I*pi*x*
*4/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 12870*x**3*sqrt(1 - x)/(1920*x**7 - 5760*x**6 + 5760*x**5
- 1920*x**4) + 2860*x**2*sqrt(1 - x)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 1040*x*sqrt(1 - x)/(19
20*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4) + 480*sqrt(1 - x)/(1920*x**7 - 5760*x**6 + 5760*x**5 - 1920*x**4)
, True))

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Giac [A]  time = 1.06646, size = 140, normalized size = 1.19 \begin{align*} \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} \, \log \left (\sqrt{-x + 1} + 1\right ) + \frac{3003}{128} \, \log \left ({\left | \sqrt{-x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(7/2)/x^5,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*log(sqrt(-x + 1) + 1) + 3003/128*lo
g(abs(sqrt(-x + 1) - 1))