3.5.37 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^5} \, dx\) [437]

Optimal. Leaf size=138 \[ -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \]

[Out]

-2/5*(a+1/x)/a^2/c^5/(1-1/a^2/x^2)^(5/2)+1/15*(-10*a-13/x)/a^2/c^5/(1-1/a^2/x^2)^(3/2)+2*arctanh((1-1/a^2/x^2)
^(1/2))/a/c^5+1/15*(-30*a-41/x)/a^2/c^5/(1-1/a^2/x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^5

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Rubi [A]
time = 0.27, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \begin {gather*} -\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^5}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]

[Out]

(-2*(a + x^(-1)))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(5/2)) - (10*a + 13/x)/(15*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2)) - (
30*a + 41/x)/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^5 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x
^2)]])/(a*c^5)

Rule 65

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 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 6312

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c + d*x)^(p -
n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)
/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (c-\frac {c x}{a}\right )^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^2}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^2-\frac {10 c^2 x}{a}-\frac {8 c^2 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^7}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^2+\frac {30 c^2 x}{a}+\frac {26 c^2 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 c^7}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^2-\frac {30 c^2 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^5}\\ &=-\frac {2 \left (a+\frac {1}{x}\right )}{5 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {10 a+\frac {13}{x}}{15 a^2 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {30 a+\frac {41}{x}}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^5}+\frac {2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^5}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 104, normalized size = 0.75 \begin {gather*} \frac {-56+82 a x+32 a^2 x^2-76 a^3 x^3+15 a^4 x^4+30 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^5 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]

[Out]

(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[
1 - 1/(a^2*x^2)]])/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(614\) vs. \(2(122)=244\).
time = 0.16, size = 615, normalized size = 4.46

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{5}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{5} \sqrt {a^{2}}}-\frac {383 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{120 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}-\frac {41 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}\right ) a^{5} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{5} \left (a x -1\right )}\)	\(259\)
default	\(-\frac {\left (-75 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+45 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+150 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+2 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+75 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-64 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-300 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-14 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x +75 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+37 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+150 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +120 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -75 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-60 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{30 a \sqrt {a^{2}}\, c^{5} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )^{5}}\)	\(615\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x,method=_RETURNVERBOSE)

[Out]

-1/30*(-75*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^6*x^6-60*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)
^(1/2))*a^7*x^6+45*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)*a^4*x^4+150*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^5*x^5
+120*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^6*x^5+2*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)
*a^3*x^3+75*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^4*x^4+60*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2
)^(1/2))*a^5*x^4-64*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*a^2*x^2-300*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^
3-240*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3-14*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/
2)*a*x+75*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2+60*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^
(1/2))*a^3*x^2+37*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)+150*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a*x+120*ln((a^2*
x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^2*x-75*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)-60*a*ln((a^2*
x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2)))/a*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/c^5/((a*x+1)*(a*x-1
))^(1/2)/(a*x-1)^5

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Maxima [A]
time = 0.27, size = 176, normalized size = 1.28 \begin {gather*} \frac {1}{120} \, a {\left (\frac {\frac {32 \, {\left (a x - 1\right )}}{a x + 1} + \frac {310 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {585 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{5}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{5}} + \frac {15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="maxima")

[Out]

1/120*a*((32*(a*x - 1)/(a*x + 1) + 310*(a*x - 1)^2/(a*x + 1)^2 - 585*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^5*((a
*x - 1)/(a*x + 1))^(7/2) - a^2*c^5*((a*x - 1)/(a*x + 1))^(5/2)) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*
c^5) - 240*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^5) + 15*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^5))

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Fricas [A]
time = 0.34, size = 170, normalized size = 1.23 \begin {gather*} \frac {30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="fricas")

[Out]

1/15*(30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 76*a^3*x^3 + 32*a^2*x^2 + 82*a*x - 56)*sqrt((a*x - 1
)/(a*x + 1)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{5} \left (\int \left (- \frac {x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\right )\, dx + \int \frac {a x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\, dx\right )}{c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**5,x)

[Out]

a**5*(Integral(-x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 4*a**5*x**5 + 5*a**4*x**4 - 5*a**2*x**2 +
4*a*x - 1), x) + Integral(a*x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 4*a**5*x**5 + 5*a**4*x**4 - 5*
a**2*x**2 + 4*a*x - 1), x))/c**5

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 0.09, size = 144, normalized size = 1.04 \begin {gather*} \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^5}-\frac {\frac {62\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {39\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {32\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^5,x)

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(8*a*c^5) - ((62*(a*x - 1)^2)/(3*(a*x + 1)^2) - (39*(a*x - 1)^3)/(a*x + 1)^3 + (32
*(a*x - 1))/(15*(a*x + 1)) + 1/5)/(8*a*c^5*((a*x - 1)/(a*x + 1))^(5/2) - 8*a*c^5*((a*x - 1)/(a*x + 1))^(7/2))
+ (4*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^5)

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