3.690 \(\int \frac {\sqrt {a+b \sin ^{-1}(c x)}}{(d+e x^2)^2} \, dx\)
Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sqrt {a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2},x\right ) \]
[Out]
Unintegrable((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x)
________________________________________________________________________________________
Rubi [A] time = 0.05, 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 =
{} \[ \int \frac {\sqrt {a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2,x]
[Out]
Defer[Int][Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx &=\int \frac {\sqrt {a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 23.49, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2,x]
[Out]
Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^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((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (co
nstant residues)
________________________________________________________________________________________
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+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.19Unable to divide, perhaps due to rounding error%%%{68719476736,[0,8,8,24,68,8,36,18]%%%}+%%%{
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%{-3199750635520,[0,6,6,24,58,10,38,19]%%%}+%%%{-18249316040704,[0,6,6,24,56,10,40,20]%%%}+%%%{-77670188580864
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%%{-8370891259904,[0,6,6,24,26,10,70,35]%%%}+%%%{-953482739712,[0,6,6,24,24,10,72,36]%%%}+%%%{-51539607552,[0,
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{41710172241920,[0,4,12,24,50,8,38,19]%%%}+%%%{103946613948416,[0,4,12,24,48,8,40,20]%%%}+%%%{213205632679936,
[0,4,12,24,46,8,42,21]%%%}+%%%{363340241043456,[0,4,12,24,44,8,44,22]%%%}+%%%{517463699619840,[0,4,12,24,42,8,
46,23]%%%}+%%%{617529978388480,[0,4,12,24,40,8,48,24]%%%}+%%%{617336285429760,[0,4,12,24,38,8,50,25]%%%}+%%%{5
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30]%%%}+%%%{9126805504000,[0,4,12,24,26,8,62,31]%%%}+%%%{1775163670528,[0,4,12,24,24,8,64,32]%%%}+%%%{21904333
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4,52,9,36,18]%%%}+%%%{45256137506816*i,[0,4,10,24,50,9,38,19]%%%}+%%%{124129554464768*i,[0,4,10,24,48,9,40,20]
%%%}+%%%{277062535348224*i,[0,4,10,24,46,9,42,21]%%%}+%%%{508893763469312*i,[0,4,10,24,44,9,44,22]%%%}+%%%{774
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24,26,8,54,27]%%%}+%%%{148394475520,[0,2,14,24,24,8,56,28]%%%}+%%%{31037849600,[0,2,14,24,22,8,58,29]%%%}+%%%{
4160749568,[0,2,14,24,20,8,60,30]%%%}+%%%{268435456,[0,2,14,24,18,8,62,31]%%%}+%%%{1048576*i,[0,2,12,24,60,9,2
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i,[0,2,12,24,54,9,26,13]%%%}+%%%{33929822208*i,[0,2,12,24,52,9,28,14]%%%}+%%%{160738312192*i,[0,2,12,24,50,9,3
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,18,2,10,5]%%%}+%%%{-6656,[0,2,2,4,16,2,12,6]%%%}+%%%{-11264,[0,2,2,4,14,2,14,7]%%%}+%%%{-10496,[0,2,2,4,12,2,
16,8]%%%}+%%%{-5120,[0,2,2,4,10,2,18,9]%%%}+%%%{-1024,[0,2,2,4,8,2,20,10]%%%}+%%%{-12,[0,0,4,4,16,2,4,2]%%%}+%
%%{-80,[0,0,4,4,14,2,6,3]%%%}+%%%{-220,[0,0,4,4,12,2,8,4]%%%}+%%%{-312,[0,0,4,4,10,2,10,5]%%%}+%%%{-224,[0,0,4
,4,8,2,12,6]%%%}+%%%{-64,[0,0,4,4,6,2,14,7]%%%}+%%%{-8*i,[0,0,2,4,16,3,4,2]%%%}+%%%{-64*i,[0,0,2,4,14,3,6,3]%%
%}+%%%{-200*i,[0,0,2,4,12,3,8,4]%%%}+%%%{-304*i,[0,0,2,4,10,3,10,5]%%%}+%%%{-224*i,[0,0,2,4,8,3,12,6]%%%}+%%%{
-64*i,[0,0,2,4,6,3,14,7]%%%}+%%%{1,[0,0,0,4,16,4,4,2]%%%}+%%%{12,[0,0,0,4,14,4,6,3]%%%}+%%%{45,[0,0,0,4,12,4,8
,4]%%%}+%%%{74,[0,0,0,4,10,4,10,5]%%%}+%%%{56,[0,0,0,4,8,4,12,6]%%%}+%%%{16,[0,0,0,4,6,4,14,7]%%%} Error: Bad
Argument Value
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maple [A] time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +b \arcsin \left (c x \right )}}{\left (e \,x^{2}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x)
[Out]
int((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x)
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \arcsin \left (c x\right ) + a}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="maxima")
[Out]
integrate(sqrt(b*arcsin(c*x) + a)/(e*x^2 + d)^2, x)
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*asin(c*x))^(1/2)/(d + e*x^2)^2,x)
[Out]
int((a + b*asin(c*x))^(1/2)/(d + e*x^2)^2, x)
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asin(c*x))**(1/2)/(e*x**2+d)**2,x)
[Out]
Integral(sqrt(a + b*asin(c*x))/(d + e*x**2)**2, x)
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