3.487 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\)

Optimal. Leaf size=199 \[ \frac {x \left (1-\frac {1}{a x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{\sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]

[Out]

-(1-1/a/x)^(5/2)*arctanh((1+1/a/x)^(1/2))/a/(c-c/a/x)^(5/2)-1/2*(1-1/a/x)^(5/2)*arctanh(1/2*(1+1/a/x)^(1/2)*2^
(1/2))/a/(c-c/a/x)^(5/2)*2^(1/2)+2*(1-1/a/x)^(5/2)/a/(c-c/a/x)^(5/2)/(1+1/a/x)^(1/2)+(1-1/a/x)^(5/2)*x/(c-c/a/
x)^(5/2)/(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6182, 6179, 103, 152, 156, 63, 208, 206} \[ \frac {x \left (1-\frac {1}{a x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{\sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]

[Out]

(2*(1 - 1/(a*x))^(5/2))/(a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2)) + ((1 - 1/(a*x))^(5/2)*x)/(Sqrt[1 + 1/(a*x)]
*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*
x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(Sqrt[2]*a*(c - c/(a*x))^(5/2))

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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])

Rule 208

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

Rule 6179

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

Rule 6182

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

Rubi steps

\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=\frac {\left (1-\frac {1}{a x}\right )^{5/2} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {\frac {1}{2 a}-\frac {3 x}{2 a^2}}{x \left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2 a^2}-\frac {x}{a^3}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1-\frac {1}{a x}\right )^{5/2}}{a \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\left (1-\frac {1}{a x}\right )^{5/2} x}{\sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{\sqrt {2} a \left (c-\frac {c}{a x}\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 90, normalized size = 0.45 \[ \frac {\sqrt {1-\frac {1}{a x}} \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+\frac {1}{x}}{2 a}\right )+\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{a x}\right )+a x\right )}{a c^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]

[Out]

(Sqrt[1 - 1/(a*x)]*(a*x + Hypergeometric2F1[-1/2, 1, 1/2, (a + x^(-1))/(2*a)] + Hypergeometric2F1[-1/2, 1, 1/2
, 1 + 1/(a*x)]))/(a*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])

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fricas [A]  time = 0.69, size = 524, normalized size = 2.63 \[ \left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(sqrt(2)*(a*x - 1)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2
+ a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*
(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x +
1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 8*(a^2*x^2 + 2*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(
a*x)))/(a^2*c^3*x - a*c^3), 1/4*(sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*
x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + 2*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^
2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(a^2*x^2 +
2*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*c^3*x - a*c^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is re
al):Check [abs(t_nostep)]Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warnin
g, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a subst
itution variable should perhaps be purged.Warning, integration of abs or sign assumes constant sign by interva
ls (correct if the argument is real):Check [abs(t_nostep)]Warning, replacing 0 by ` u`, a substitution variabl
e should perhaps be purged.Warning, integration of abs or sign assumes constant sign by intervals (correct if
the argument is real):Check [abs(t_nostep)]Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}
+%%%{2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{3,[2,2,2]%%%}+%%%{-2,[1,2,1]%%%}+%%%{1,[0,2,0]%%
%}] at parameters values [86,-97,-82]Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{2
,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{3,[2,2,2]%%%}+%%%{-2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at
parameters values [-89,63,-49]Warning, choosing root of [1,0,%%%{2,[1,1]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[2,2]%%%
}+%%%{-2,[1,2]%%%}+%%%{1,[0,2]%%%}] at parameters values [-64,-15.6438432182]Warning, choosing root of [1,0,%%
%{2,[1,1]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[2,2]%%%}+%%%{-2,[1,2]%%%}+%%%{1,[0,2]%%%}] at parameters values [42,-55
.0901457258]Warning, choosing root of [1,0,%%%{2,[1,1]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[2,2]%%%}+%%%{-2,[1,2]%%%}+
%%%{1,[0,2]%%%}] at parameters values [46,-26.2290649475]Undef/Unsigned Inf encountered in limitEvaluation tim
e: 0.42Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.07, size = 262, normalized size = 1.32 \[ -\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-4 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, x +a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x +2 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -8 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}+2 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{4 \left (a x -1\right )^{2} a^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)/c^3*(-4*a^(5/2)*(1/a)^(1/2)*((a
*x+1)*x)^(1/2)*x+a^(3/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x+2*ln(1/2*(2
*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x-8*((a*x+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)+2^(1/2)
*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*a^(1/2)+2*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)
+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2))/(1/a)^(1/2)/((a*x+1)*x)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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