3.518 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^{9/2} \, dx\)
Optimal. Leaf size=145 \[ \frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}-\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}-\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}-x \left (c-\frac {c}{a x}\right )^{9/2} \]
[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 = 0.19, antiderivative size = 145, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used
= {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*ArcTanh[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]
Rubi steps
\begin {align*} \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 = 0.15, size = 91, normalized size = 0.63 \[ \frac {5 c^{9/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {c^4 \left (21 a^4 x^4+92 a^3 x^3+4 a^2 x^2-18 a x+6\right ) \sqrt {c-\frac {c}{a x}}}{21 a^4 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[E^(2*ArcTanh[a*x])*(c - c/(a*x))^(9/2),x]
[Out]
-1/21*(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))/(a^4*x^3) + (5*c^(9/2)*ArcTan
h[Sqrt[c - c/(a*x)]/Sqrt[c]])/a
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fricas [A] time = 0.46, size = 234, normalized size = 1.61 \[ \left [\frac {105 \, a^{3} c^{\frac {9}{2}} x^{3} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{42 \, a^{4} x^{3}}, -\frac {105 \, a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{21 \, a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+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((a*x+1)^2/(-a^2*x^2+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: 1.79Limit: 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 = 1.12 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (-210 a^{\frac {9}{2}} \sqrt {a \,x^{2}-x}\, x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{5} a^{4}+168 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{3}-16 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-24 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{42 x^{4} \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(9/2),x)
[Out]
1/42*(c*(a*x-1)/a/x)^(1/2)/x^4*c^4*(-210*a^(9/2)*(a*x^2-x)^(1/2)*x^5+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+168*a^(7/2)*(a*x^2-x)^(3/2)*x^3-16*a^(5/2)*(a*x^2-x)^(3/2)*x^2-24*a^(3/2)*(a*x^2-x)^(3/
2)*x+12*(a*x^2-x)^(3/2)*a^(1/2))/((a*x-1)*x)^(1/2)/a^(9/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(9/2),x, algorithm="maxima")
[Out]
-integrate((a*x + 1)^2*(c - c/(a*x))^(9/2)/(a^2*x^2 - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((c - c/(a*x))^(9/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
[Out]
int(-((c - c/(a*x))^(9/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)
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sympy [C] time = 31.19, size = 2421, normalized size = 16.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(9/2),x)
[Out]
-c**4*Piecewise((sqrt(a)*sqrt(c)*x**(3/2)/sqrt(a*x - 1) - sqrt(c)*acosh(sqrt(a)*sqrt(x))/a - sqrt(c)*sqrt(x)/(
sqrt(a)*sqrt(a*x - 1)), Abs(a*x) > 1), (I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)*sqrt(-a*x + 1)/s
qrt(a), True)) - 4*c**5*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 4*c**4*sqrt(c - c/(a*x))/a - 2*c**4*Pi
ecewise((4*I*a**(11/2)*sqrt(c)*x**(7/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**(9/2)*sqrt
(c)*x**(5/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**5*sqrt(c)*x**3*sqrt(a*x - 1)/(-15*I*a
**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) + 2*I*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15
*I*a**(5/2)*x**(5/2)) + 8*I*a**3*sqrt(c)*x*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) -
6*I*a**2*sqrt(c)*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*I*a**(11
/2)*sqrt(c)*x**(7/2)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 4*I*a**(9/2)*sqrt(c)*x**(5/2)/(15*I*a
**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 4*a**5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I
*a**(5/2)*x**(5/2)) + 2*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 8
*a**3*sqrt(c)*x*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 6*a**2*sqrt(c)*sqrt(-a*x +
1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)), True))/a**3 + c**4*Piecewise((16*I*a**(19/2)*sqrt(c)*x**
(13/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(
7/2)) - 48*I*a**(17/2)*sqrt(c)*x**(11/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9
/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 48*I*a**(15/2)*sqrt(c)*x**(9/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*
a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16*I*a**(13/2)*sqrt(c)*x**(7/2)/(-1
05*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16
*I*a**9*sqrt(c)*x**6*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x*
*(9/2) + 105*I*a**(7/2)*x**(7/2)) + 40*I*a**8*sqrt(c)*x**5*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a
**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 30*I*a**7*sqrt(c)*x**4*sqrt(a*x - 1)
/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2))
+ 40*I*a**6*sqrt(c)*x**3*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2
)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 100*I*a**5*sqrt(c)*x**2*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 31
5*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 96*I*a**4*sqrt(c)*x*sqrt(a*x -
1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)
) - 30*I*a**3*sqrt(c)*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x
**(9/2) + 105*I*a**(7/2)*x**(7/2)), Abs(a*x) > 1), (16*I*a**(19/2)*sqrt(c)*x**(13/2)/(-105*I*a**(13/2)*x**(13/
2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 48*I*a**(17/2)*sqrt(c)*x
**(11/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x*
*(7/2)) + 48*I*a**(15/2)*sqrt(c)*x**(9/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(
9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16*I*a**(13/2)*sqrt(c)*x**(7/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I
*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 16*a**9*sqrt(c)*x**6*sqrt(-a*x + 1
)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2))
- 40*a**8*sqrt(c)*x**5*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2
)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 30*a**7*sqrt(c)*x**4*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*
I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 40*a**6*sqrt(c)*x**3*sqrt(-a*x +
1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)
) + 100*a**5*sqrt(c)*x**2*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9
/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 96*a**4*sqrt(c)*x*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I
*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 30*a**3*sqrt(c)*sqrt(-a*x + 1)/(-1
05*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)), Tru
e))/a**4
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