3.519 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\)
Optimal. Leaf size=120 \[ \frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-x \left (c-\frac {c}{a x}\right )^{7/2} \]
[Out]
-c^2*(c-c/a/x)^(3/2)/a-3/5*c*(c-c/a/x)^(5/2)/a-(c-c/a/x)^(7/2)*x+3*c^(7/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a-
3*c^3*(c-c/a/x)^(1/2)/a
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Rubi [A] time = 0.16, antiderivative size = 120, normalized size of antiderivative = 1.00,
number of steps used = 10, 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} \[ -\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}+\frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-x \left (c-\frac {c}{a x}\right )^{7/2} \]
Antiderivative was successfully verified.
[In]
Int[E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2),x]
[Out]
(-3*c^3*Sqrt[c - c/(a*x)])/a - (c^2*(c - c/(a*x))^(3/2))/a - (3*c*(c - c/(a*x))^(5/2))/(5*a) - (c - c/(a*x))^(
7/2)*x + (3*c^(7/2)*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 )^{7/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)}{1-a x} \, dx\\ &=-\frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{5/2} (1+a x)}{x} \, dx}{a}\\ &=-\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {(3 c) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^2\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 {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.69 \[ \frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {c^3 \left (5 a^3 x^3+8 a^2 x^2+4 a x-2\right ) \sqrt {c-\frac {c}{a x}}}{5 a^3 x^2} \]
Antiderivative was successfully verified.
[In]
Integrate[E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2),x]
[Out]
-1/5*(c^3*Sqrt[c - c/(a*x)]*(-2 + 4*a*x + 8*a^2*x^2 + 5*a^3*x^3))/(a^3*x^2) + (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a
*x)]/Sqrt[c]])/a
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fricas [A] time = 0.41, size = 212, normalized size = 1.77 \[ \left [\frac {15 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, a^{3} x^{2}}, -\frac {15 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, a^{3} x^{2}}\right ] \]
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)^(7/2),x, algorithm="fricas")
[Out]
[1/10*(15*a^2*c^(7/2)*x^2*log(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) - 2*(5*a^3*c^3*x^3 + 8*a^2
*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2), -1/5*(15*a^2*sqrt(-c)*c^3*x^2*arctan(sqrt(-c
)*sqrt((a*c*x - c)/(a*x))/c) + (5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a*c*x - c)/(a*x)))/(a
^3*x^2)]
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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)^(7/2),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,[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 parameters values [76,-66,66]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 [5,-23,79]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 [-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,[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 [-92,80,-24]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 [-17,41,64]Sign e
rror (%%%{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%%%})Limit: Max order reached or unable to make series
expansion Error: Bad Argument Value
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maple [A] time = 0.04, size = 144, normalized size = 1.20 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (-30 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x^{4}+20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{4} a^{3}+4 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{10 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}} \]
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)^(7/2),x)
[Out]
1/10*(c*(a*x-1)/a/x)^(1/2)/x^3*c^3*(-30*(a*x^2-x)^(1/2)*a^(7/2)*x^4+20*a^(5/2)*(a*x^2-x)^(3/2)*x^2+15*ln(1/2*(
2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*x^4*a^3+4*a^(3/2)*(a*x^2-x)^(3/2)*x-4*(a*x^2-x)^(3/2)*a^(1/2))/((a
*x-1)*x)^(1/2)/a^(7/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{a^{2} x^{2} - 1}\,{d x} \]
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)^(7/2),x, algorithm="maxima")
[Out]
-integrate((a*x + 1)^2*(c - c/(a*x))^(7/2)/(a^2*x^2 - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((c - c/(a*x))^(7/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
[Out]
int(-((c - c/(a*x))^(7/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)
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sympy [C] time = 25.99, size = 777, normalized size = 6.48 \[ - c^{3} \left (\begin {cases} \frac {\sqrt {a} \sqrt {c} x^{\frac {3}{2}}}{\sqrt {a x - 1}} - \frac {\sqrt {c} \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} - \frac {\sqrt {c} \sqrt {x}}{\sqrt {a} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {i \sqrt {c} \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {i \sqrt {c} \sqrt {x} \sqrt {- a x + 1}}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {2 c^{4} \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 c^{3} \sqrt {c - \frac {c}{a x}}}{a} + \frac {c^{3} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {2 a \left (c - \frac {c}{a x}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {c^{3} \left (\begin {cases} \frac {4 i a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {4 i a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {4 i a^{5} \sqrt {c} x^{3} \sqrt {a x - 1}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {2 i a^{4} \sqrt {c} x^{2} \sqrt {a x - 1}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {8 i a^{3} \sqrt {c} x \sqrt {a x - 1}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {6 i a^{2} \sqrt {c} \sqrt {a x - 1}}{- 15 i a^{\frac {7}{2}} x^{\frac {7}{2}} + 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {4 i a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 i a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {4 a^{5} \sqrt {c} x^{3} \sqrt {- a x + 1}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {2 a^{4} \sqrt {c} x^{2} \sqrt {- a x + 1}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {8 a^{3} \sqrt {c} x \sqrt {- a x + 1}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {6 a^{2} \sqrt {c} \sqrt {- a x + 1}}{15 i a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 i a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right )}{a^{3}} \]
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)**(7/2),x)
[Out]
-c**3*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)) - 2*c**4*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 2*c**3*sqrt(c - c/(a*x))/a + c**3*Piec
ewise((0, Eq(c, 0)), (2*a*(c - c/(a*x))**(3/2)/(3*c), True))/a**2 - c**3*Piecewise((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*sq
rt(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
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