3.703 \(\int \frac {1}{(d+e x^2)^2 (a+b \sin ^{-1}(c x))^{3/2}} \, dx\)
Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}},x\right ) \]
[Out]
Unintegrable(1/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x)
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)),x]
[Out]
Defer[Int][1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)), x]
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=\int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)),x]
[Out]
Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)), x]
________________________________________________________________________________________
fricas [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/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (co
nstant residues)
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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/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.48Unable to divide, perhaps due to rounding error%%%{68719476736,[0,8,72,24,52,8,36,18]%%%}+%%%
{687194767360,[0,8,72,24,50,8,38,19]%%%}+%%%{2817498546176,[0,8,72,24,48,8,40,20]%%%}+%%%{6047313952768,[0,8,7
2,24,46,8,42,21]%%%}+%%%{7146825580544,[0,8,72,24,44,8,44,22]%%%}+%%%{4398046511104,[0,8,72,24,42,8,46,23]%%%}
+%%%{1099511627776,[0,8,72,24,40,8,48,24]%%%}+%%%{-2147483648,[0,6,66,24,50,8,30,15]%%%}+%%%{27917287424,[0,6,
66,24,48,8,32,16]%%%}+%%%{135291469824,[0,6,66,24,46,8,34,17]%%%}+%%%{324270030848,[0,6,66,24,44,8,36,18]%%%}+
%%%{412316860416,[0,6,66,24,42,8,38,19]%%%}+%%%{266287972352,[0,6,66,24,40,8,40,20]%%%}+%%%{68719476736,[0,6,6
6,24,38,8,42,21]%%%}+%%%{-12884901888*i,[0,6,64,24,48,9,32,16]%%%}+%%%{-103079215104*i,[0,6,64,24,46,9,34,17]%
%%}+%%%{-322122547200*i,[0,6,64,24,44,9,36,18]%%%}+%%%{-489626271744*i,[0,6,64,24,42,9,38,19]%%%}+%%%{-3607772
52864*i,[0,6,64,24,40,9,40,20]%%%}+%%%{-103079215104*i,[0,6,64,24,38,9,42,21]%%%}+%%%{-9663676416,[0,6,62,24,4
8,10,32,16]%%%}+%%%{-91804925952,[0,6,62,24,46,10,34,17]%%%}+%%%{-333396836352,[0,6,62,24,44,10,36,18]%%%}+%%%
{-579820584960,[0,6,62,24,42,10,38,19]%%%}+%%%{-483183820800,[0,6,62,24,40,10,40,20]%%%}+%%%{-154618822656,[0,
6,62,24,38,10,42,21]%%%}+%%%{16777216,[0,4,60,24,48,8,24,12]%%%}+%%%{369098752,[0,4,60,24,46,8,26,13]%%%}+%%%{
2315255808,[0,4,60,24,44,8,28,14]%%%}+%%%{6408896512,[0,4,60,24,42,8,30,15]%%%}+%%%{8875147264,[0,4,60,24,40,8
,32,16]%%%}+%%%{6039797760,[0,4,60,24,38,8,34,17]%%%}+%%%{1610612736,[0,4,60,24,36,8,36,18]%%%}+%%%{-402653184
*i,[0,4,58,24,46,9,26,13]%%%}+%%%{-3825205248*i,[0,4,58,24,44,9,28,14]%%%}+%%%{-13287555072*i,[0,4,58,24,42,9,
30,15]%%%}+%%%{-21541945344*i,[0,4,58,24,40,9,32,16]%%%}+%%%{-16508780544*i,[0,4,58,24,38,9,34,17]%%%}+%%%{-48
31838208*i,[0,4,58,24,36,9,36,18]%%%}+%%%{-75497472,[0,4,56,24,46,10,26,13]%%%}+%%%{-2113929216,[0,4,56,24,44,
10,28,14]%%%}+%%%{-11324620800,[0,4,56,24,42,10,30,15]%%%}+%%%{-23781703680,[0,4,56,24,40,10,32,16]%%%}+%%%{-2
1743271936,[0,4,56,24,38,10,34,17]%%%}+%%%{-7247757312,[0,4,56,24,36,10,36,18]%%%}+%%%{1358954496*i,[0,4,54,24
,44,11,28,14]%%%}+%%%{9965666304*i,[0,4,54,24,42,11,30,15]%%%}+%%%{25820135424*i,[0,4,54,24,40,11,32,16]%%%}+%
%%{28085059584*i,[0,4,54,24,38,11,34,17]%%%}+%%%{10871635968*i,[0,4,54,24,36,11,36,18]%%%}+%%%{509607936,[0,4,
52,24,44,12,28,14]%%%}+%%%{4586471424,[0,4,52,24,42,12,30,15]%%%}+%%%{14353956864,[0,4,52,24,40,12,32,16]%%%}+
%%%{18345885696,[0,4,52,24,38,12,34,17]%%%}+%%%{8153726976,[0,4,52,24,36,12,36,18]%%%}+%%%{1572864,[0,2,54,24,
44,8,20,10]%%%}+%%%{16252928,[0,2,54,24,42,8,22,11]%%%}+%%%{55050240,[0,2,54,24,40,8,24,12]%%%}+%%%{84410368,[
0,2,54,24,38,8,26,13]%%%}+%%%{60817408,[0,2,54,24,36,8,28,14]%%%}+%%%{16777216,[0,2,54,24,34,8,30,15]%%%}+%%%{
-3145728*i,[0,2,52,24,44,9,20,10]%%%}+%%%{-44040192*i,[0,2,52,24,42,9,22,11]%%%}+%%%{-179306496*i,[0,2,52,24,4
0,9,24,12]%%%}+%%%{-314572800*i,[0,2,52,24,38,9,26,13]%%%}+%%%{-251658240*i,[0,2,52,24,36,9,28,14]%%%}+%%%{-75
497472*i,[0,2,52,24,34,9,30,15]%%%}+%%%{-1179648,[0,2,50,24,44,10,20,10]%%%}+%%%{-30670848,[0,2,50,24,42,10,22
,11]%%%}+%%%{-174587904,[0,2,50,24,40,10,24,12]%%%}+%%%{-376307712,[0,2,50,24,38,10,26,13]%%%}+%%%{-344457216,
[0,2,50,24,36,10,28,14]%%%}+%%%{-113246208,[0,2,50,24,34,10,30,15]%%%}+%%%{56623104*i,[0,2,48,24,40,11,24,12]%
%%}+%%%{226492416*i,[0,2,48,24,38,11,26,13]%%%}+%%%{283115520*i,[0,2,48,24,36,11,28,14]%%%}+%%%{113246208*i,[0
,2,48,24,34,11,30,15]%%%}+%%%{-2654208,[0,2,46,24,42,12,22,11]%%%}+%%%{45121536,[0,2,46,24,40,12,24,12]%%%}+%%
%{294617088,[0,2,46,24,38,12,26,13]%%%}+%%%{498991104,[0,2,46,24,36,12,28,14]%%%}+%%%{254803968,[0,2,46,24,34,
12,30,15]%%%}+%%%{-47775744*i,[0,2,44,24,40,13,24,12]%%%}+%%%{-318504960*i,[0,2,44,24,38,13,26,13]%%%}+%%%{-63
7009920*i,[0,2,44,24,36,13,28,14]%%%}+%%%{-382205952*i,[0,2,44,24,34,13,30,15]%%%}+%%%{-11943936,[0,2,42,24,40
,14,24,12]%%%}+%%%{-101523456,[0,2,42,24,38,14,26,13]%%%}+%%%{-262766592,[0,2,42,24,36,14,28,14]%%%}+%%%{-1911
02976,[0,2,42,24,34,14,30,15]%%%}+%%%{36864,[0,0,48,24,40,8,16,8]%%%}+%%%{172032,[0,0,48,24,38,8,18,9]%%%}+%%%
{299008,[0,0,48,24,36,8,20,10]%%%}+%%%{229376,[0,0,48,24,34,8,22,11]%%%}+%%%{65536,[0,0,48,24,32,8,24,12]%%%}+
%%%{-147456*i,[0,0,46,24,40,9,16,8]%%%}+%%%{-786432*i,[0,0,46,24,38,9,18,9]%%%}+%%%{-1523712*i,[0,0,46,24,36,9
,20,10]%%%}+%%%{-1277952*i,[0,0,46,24,34,9,22,11]%%%}+%%%{-393216*i,[0,0,46,24,32,9,24,12]%%%}+%%%{-202752,[0,
0,44,24,40,10,16,8]%%%}+%%%{-1124352,[0,0,44,24,38,10,18,9]%%%}+%%%{-2248704,[0,0,44,24,36,10,20,10]%%%}+%%%{-
1916928,[0,0,44,24,34,10,22,11]%%%}+%%%{-589824,[0,0,44,24,32,10,24,12]%%%}+%%%{110592*i,[0,0,42,24,40,11,16,8
]%%%}+%%%{221184*i,[0,0,42,24,38,11,18,9]%%%}+%%%{-552960*i,[0,0,42,24,36,11,20,10]%%%}+%%%{-1548288*i,[0,0,42
,24,34,11,22,11]%%%}+%%%{-884736*i,[0,0,42,24,32,11,24,12]%%%}+%%%{20736,[0,0,40,24,40,12,16,8]%%%}+%%%{-70502
4,[0,0,40,24,38,12,18,9]%%%}+%%%{-4022784,[0,0,40,24,36,12,20,10]%%%}+%%%{-6635520,[0,0,40,24,34,12,22,11]%%%}
+%%%{-3317760,[0,0,40,24,32,12,24,12]%%%}+%%%{497664*i,[0,0,38,24,38,13,18,9]%%%}+%%%{2737152*i,[0,0,38,24,36,
13,20,10]%%%}+%%%{4478976*i,[0,0,38,24,34,13,22,11]%%%}+%%%{1990656*i,[0,0,38,24,32,13,24,12]%%%}+%%%{93312,[0
,0,36,24,38,14,18,9]%%%}+%%%{-186624,[0,0,36,24,36,14,20,10]%%%}+%%%{-2239488,[0,0,36,24,34,14,22,11]%%%}+%%%{
-2985984,[0,0,36,24,32,14,24,12]%%%}+%%%{559872*i,[0,0,34,24,36,15,20,10]%%%}+%%%{3359232*i,[0,0,34,24,34,15,2
2,11]%%%}+%%%{4478976*i,[0,0,34,24,32,15,24,12]%%%}+%%%{104976,[0,0,32,24,36,16,20,10]%%%}+%%%{839808,[0,0,32,
24,34,16,22,11]%%%}+%%%{1679616,[0,0,32,24,32,16,24,12]%%%} / %%%{-256,[0,2,18,4,16,2,8,4]%%%}+%%%{-1024,[0,2,
18,4,14,2,10,5]%%%}+%%%{-1024,[0,2,18,4,12,2,12,6]%%%}+%%%{-12,[0,0,12,4,12,2,4,2]%%%}+%%%{-16,[0,0,12,4,10,2,
6,3]%%%}+%%%{24*i,[0,0,10,4,12,3,4,2]%%%}+%%%{48*i,[0,0,10,4,10,3,6,3]%%%}+%%%{9,[0,0,8,4,12,4,4,2]%%%}+%%%{36
,[0,0,8,4,10,4,6,3]%%%} Error: Bad Argument Value
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maple [A] time = 1.24, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x)
[Out]
int(1/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x)
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")
[Out]
integrate(1/((e*x^2 + d)^2*(b*arcsin(c*x) + a)^(3/2)), x)
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*asin(c*x))^(3/2)*(d + e*x^2)^2),x)
[Out]
int(1/((a + b*asin(c*x))^(3/2)*(d + e*x^2)^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(1/(e*x**2+d)**2/(a+b*asin(c*x))**(3/2),x)
[Out]
Timed out
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