3.446 \(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{9/2} \, dx\)
Optimal. Leaf size=143 \[ -\frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+x \left (c-\frac {c}{a x}\right )^{9/2} \]
[Out]
5/3*c^3*(c-c/a/x)^(3/2)/a+c^2*(c-c/a/x)^(5/2)/a+5/7*c*(c-c/a/x)^(7/2)/a+(c-c/a/x)^(9/2)*x-5*c^(9/2)*arctanh((c
-c/a/x)^(1/2)/c^(1/2))/a+5*c^4*(c-c/a/x)^(1/2)/a
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Rubi [A] time = 0.24, antiderivative size = 143, normalized size of antiderivative = 1.00,
number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used
= {6167, 6133, 25, 514, 375, 78, 50, 63, 208} \[ \frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}-\frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+x \left (c-\frac {c}{a x}\right )^{9/2} \]
Antiderivative was successfully verified.
[In]
Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]
[Out]
(5*c^4*Sqrt[c - c/(a*x)])/a + (5*c^3*(c - c/(a*x))^(3/2))/(3*a) + (c^2*(c - c/(a*x))^(5/2))/a + (5*c*(c - c/(a
*x))^(7/2))/(7*a) + (c - c/(a*x))^(9/2)*x - (5*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a
Rule 25
Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] && !(IntegerQ[m] && NegQ[n])
Rule 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && 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 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
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 375
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Rule 514
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
| !IntegerQ[p])
Rule 6133
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[(u*(c + d/x)^p*(1 + a*x)^(n/
2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] &
& !GtQ[c, 0]
Rule 6167
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx\\ &=-\int \frac {\left (c-\frac {c}{a x}\right )^{9/2} (1+a x)}{1-a x} \, dx\\ &=\frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)}{x} \, dx}{a}\\ &=\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2} \, dx}{a}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {(5 c) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x-\left (5 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 91, normalized size = 0.64 \[ \frac {c^4 \left (21 a^4 x^4+92 a^3 x^3+4 a^2 x^2-18 a x+6\right ) \sqrt {c-\frac {c}{a x}}}{21 a^4 x^3}-\frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]
[Out]
(c^4*Sqrt[c - c/(a*x)]*(6 - 18*a*x + 4*a^2*x^2 + 92*a^3*x^3 + 21*a^4*x^4))/(21*a^4*x^3) - (5*c^(9/2)*ArcTanh[S
qrt[c - c/(a*x)]/Sqrt[c]])/a
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fricas [A] time = 0.70, size = 234, normalized size = 1.64 \[ \left [\frac {105 \, a^{3} c^{\frac {9}{2}} x^{3} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{42 \, a^{4} x^{3}}, \frac {105 \, a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{21 \, a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x, algorithm="fricas")
[Out]
[1/42*(105*a^3*c^(9/2)*x^3*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 2*(21*a^4*c^4*x^4 + 92*
a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3), 1/21*(105*a^3*sqrt(-c)*c
^4*x^3*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (21*a^4*c^4*x^4 + 92*a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^
4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^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(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),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(x)]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-97,36.6646
323889,7]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-89,7.7
9369851155,-49]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-
64,-30,70]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [22,42,
56]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-9,-13,46]War
ning, 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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [24,49,-6]Warning, c
hoosing 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
]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-49,-33,-70]Warning, choos
ing 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]%%%
}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [8,63,-64]Warning, choosing roo
t of [1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-
1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [2,62,-37]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]%%%}+%%%{-1,[2,2,
2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-80,-23,65]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]%%%}+%%%{-1,[2,2,2]%%%
}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-85,28,-44]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%
{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-22,93,91]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,
2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [31,-21,88]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%
%}+%%%{1,[0,2,0]%%%}] at parameters values [76,-66,66]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]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%
{1,[0,2,0]%%%}] at parameters values [5,-23,79]Warning, choosing root of [1,0,%%%{-2,[2,1,2]%%%}+%%%{-2,[2,0,2
]%%%}+%%%{2,[1,1,1]%%%}+%%%{2,[1,0,1]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[4,1,4]%%%}+%%%{1,[4,0,4]%%%}+%%%{-2,[3,2
,3]%%%}+%%%{4,[3,1,3]%%%}+%%%{-2,[3,0,3]%%%}+%%%{1,[2,2,2]%%%}+%%%{-2,[2,1,2]%%%}+%%%{1,[2,0,2]%%%}] at parame
ters values [-88,9,6]Warning, choosing root of [1,0,%%%{-2,[2,1,0]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,
%%%{1,[4,2,0]%%%}+%%%{-2,[3,2,1]%%%}+%%%{1,[2,2,2]%%%}+%%%{-2,[2,2,0]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]
at parameters values [-69,-8,31]Warning, choosing root of [1,0,%%%{-2,[2,1,0]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0
,1,0]%%%},0,%%%{1,[4,2,0]%%%}+%%%{-2,[3,2,1]%%%}+%%%{1,[2,2,2]%%%}+%%%{-2,[2,2,0]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,
[0,2,0]%%%}] at parameters values [89,2,97]Warning, choosing root of [1,0,%%%{-2,[2,1,0]%%%}+%%%{2,[1,1,1]%%%}
+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,0]%%%}+%%%{-2,[3,2,1]%%%}+%%%{1,[2,2,2]%%%}+%%%{-2,[2,2,0]%%%}+%%%{2,[1,2,1]%
%%}+%%%{1,[0,2,0]%%%}] at parameters values [-92,80,-24]Warning, choosing root of [1,0,%%%{-2,[2,1,0]%%%}+%%%{
2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,0]%%%}+%%%{-2,[3,2,1]%%%}+%%%{1,[2,2,2]%%%}+%%%{-2,[2,2,0]%%%}+%
%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-17,41,64]Warning, choosing root of [1,0,%%%{2,[1,1,
1]%%%}+%%%{-2,[0,2,1]%%%}+%%%{-2,[0,0,1]%%%},0,%%%{1,[2,2,2]%%%}+%%%{-2,[1,3,2]%%%}+%%%{2,[1,1,2]%%%}+%%%{1,[0
,4,2]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,0,2]%%%}] at parameters values [18,-51,-42]Warning, choosing root of [1,
0,%%%{2,[1,1,1]%%%}+%%%{-2,[0,2,1]%%%}+%%%{-2,[0,0,1]%%%},0,%%%{1,[2,2,2]%%%}+%%%{-2,[1,3,2]%%%}+%%%{2,[1,1,2]
%%%}+%%%{1,[0,4,2]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,0,2]%%%}] at parameters values [-65,22,-94]Sign error (%%%{
sqrt(c)*a,0%%%}+%%%{2*sqrt(-a*c)*abs(a),1/2%%%}+%%%{-2*sqrt(c)*a^2,1%%%}+%%%{-a*sqrt(-a*c)*abs(a),3/2%%%}+%%%{
-a^2*sqrt(-a*c)*abs(a)/4,5/2%%%}+%%%{undef,7/2%%%})Evaluation time: 3.43Limit: Max order reached or unable to
make series expansion Error: Bad Argument Value
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maple [A] time = 0.05, size = 163, normalized size = 1.14 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (210 a^{\frac {9}{2}} \sqrt {a \,x^{2}-x}\, x^{5}-168 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{3}+16 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-105 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{5} a^{4}+24 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{42 x^{4} \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)/(a*x-1)*(c-c/a/x)^(9/2),x)
[Out]
1/42*(c*(a*x-1)/a/x)^(1/2)*c^4*(210*a^(9/2)*(a*x^2-x)^(1/2)*x^5-168*a^(7/2)*(a*x^2-x)^(3/2)*x^3+16*a^(5/2)*(a*
x^2-x)^(3/2)*x^2-105*ln(1/2*(2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^5*a^4+24*a^(3/2)*(a*x^2-x)^(3/2)*x-
12*(a*x^2-x)^(3/2)*a^(1/2))/x^4/((a*x-1)*x)^(1/2)/a^(9/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{a x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x, algorithm="maxima")
[Out]
integrate((a*x + 1)*(c - c/(a*x))^(9/2)/(a*x - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((c - c/(a*x))^(9/2)*(a*x + 1))/(a*x - 1),x)
[Out]
int(((c - c/(a*x))^(9/2)*(a*x + 1))/(a*x - 1), x)
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sympy [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(1/(a*x-1)*(a*x+1)*(c-c/a/x)**(9/2),x)
[Out]
Exception raised: TypeError
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