∫ (a+b sin−1(cx))3∕2 ___________________________

3.694 \(\int \frac {(a+b \sin ^{-1}(c x))^{3/2}}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\left (d+e x^2\right )^2},x\right ) \]

[Out]

Unintegrable((a+b*arcsin(c*x))^(3/2)/(e*x^2+d)^2,x)

________________________________________________________________________________________

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 = {} \[ \int \frac {\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\left (d+e x^2\right )^2} \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

Int[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2,x]

[Out]

Defer[Int][(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2, x]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\left (d+e x^2\right )^2} \, dx &=\int \frac {\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\left (d+e x^2\right )^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 12.77, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\left (d+e x^2\right )^2} \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2,x]

[Out]

Integrate[(a + b*ArcSin[c*x])^(3/2)/(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))^(3/2)/(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))^(3/2)/(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: 1.35Unable to divide, perhaps due to rounding error%%%{1099511627776,[0,8,8,48,24,24,8,40]%%%}+%%

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2,32,24,38,9,29]%%%}+%%%{51733525325414400*i,[0,6,12,30,24,40,9,28]%%%}+%%%{53016938862870528*i,[0,6,12,28,24,

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3467327098060800*i,[0,6,12,18,24,52,9,22]%%%}+%%%{1080012476252160*i,[0,6,12,16,24,54,9,21]%%%}+%%%{2624654514

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0,24,30,10,33]%%%}+%%%{-4828910114045952,[0,6,10,38,24,32,10,32]%%%}+%%%{-11126370478325760,[0,6,10,36,24,34,1

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2256694746611712,[0,6,10,26,24,44,10,26]%%%}+%%%{-22459234394308608,[0,6,10,24,24,46,10,25]%%%}+%%%{-128966593

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%%%{-270582939648,[0,6,10,8,24,62,10,17]%%%}+%%%{-9663676416,[0,6,10,6,24,64,10,16]%%%}+%%%{12884901888,[0,4,2

0,48,24,20,8,34]%%%}+%%%{219043332096,[0,4,20,46,24,22,8,33]%%%}+%%%{1775163670528,[0,4,20,44,24,24,8,32]%%%}+

%%%{9126805504000,[0,4,20,42,24,26,8,31]%%%}+%%%{33417999679488,[0,4,20,40,24,28,8,30]%%%}+%%%{92717673611264,

[0,4,20,38,24,30,8,29]%%%}+%%%{202452846510080,[0,4,20,36,24,32,8,28]%%%}+%%%{356623079964672,[0,4,20,34,24,34

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17529978388480,[0,4,20,28,24,40,8,24]%%%}+%%%{517463699619840,[0,4,20,26,24,42,8,23]%%%}+%%%{363340241043456,[

0,4,20,24,24,44,8,22]%%%}+%%%{213205632679936,[0,4,20,22,24,46,8,21]%%%}+%%%{103946613948416,[0,4,20,20,24,48,

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477552128,[0,4,20,14,24,54,8,17]%%%}+%%%{703183454208,[0,4,20,12,24,56,8,16]%%%}+%%%{103750303744,[0,4,20,10,2

4,58,8,15]%%%}+%%%{10502537216,[0,4,20,8,24,60,8,14]%%%}+%%%{637534208,[0,4,20,6,24,62,8,13]%%%}+%%%{16777216,

[0,4,20,4,24,64,8,12]%%%}+%%%{77309411328*i,[0,4,18,48,24,20,9,34]%%%}+%%%{1314259992576*i,[0,4,18,46,24,22,9,

33]%%%}+%%%{10609106092032*i,[0,4,18,44,24,24,9,32]%%%}+%%%{54090818125824*i,[0,4,18,42,24,26,9,31]%%%}+%%%{19

5440607756288*i,[0,4,18,40,24,28,9,30]%%%}+%%%{532221341466624*i,[0,4,18,38,24,30,9,29]%%%}+%%%{11340488355348

48*i,[0,4,18,36,24,32,9,28]%%%}+%%%{1937380491657216*i,[0,4,18,34,24,34,9,27]%%%}+%%%{2697110143107072*i,[0,4,

18,32,24,36,9,26]%%%}+%%%{3092580698947584*i,[0,4,18,30,24,38,9,25]%%%}+%%%{2939655032930304*i,[0,4,18,28,24,4

0,9,24]%%%}+%%%{2323428459675648*i,[0,4,18,26,24,42,9,23]%%%}+%%%{1526681290407936*i,[0,4,18,24,24,44,9,22]%%%

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412520448*i,[0,4,18,18,24,50,9,19]%%%}+%%%{39609597689856*i,[0,4,18,16,24,52,9,18]%%%}+%%%{9018122698752*i,[0,

4,18,14,24,54,9,17]%%%}+%%%{1542061031424*i,[0,4,18,12,24,56,9,16]%%%}+%%%{186025771008*i,[0,4,18,10,24,58,9,1

5]%%%}+%%%{14092861440*i,[0,4,18,8,24,60,9,14]%%%}+%%%{503316480*i,[0,4,18,6,24,62,9,13]%%%}+%%%{-173946175488

,[0,4,16,48,24,20,10,34]%%%}+%%%{-2957084983296,[0,4,16,46,24,22,10,33]%%%}+%%%{-23805258891264,[0,4,16,44,24,

24,10,32]%%%}+%%%{-120660663730176,[0,4,16,42,24,26,10,31]%%%}+%%%{-431863961223168,[0,4,16,40,24,28,10,30]%%%

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1333866848256,[0,4,16,34,24,34,10,27]%%%}+%%%{-5476973392429056,[0,4,16,32,24,36,10,26]%%%}+%%%{-6057839919366

144,[0,4,16,30,24,38,10,25]%%%}+%%%{-5512622166245376,[0,4,16,28,24,40,10,24]%%%}+%%%{-4135295964413952,[0,4,1

6,26,24,42,10,23]%%%}+%%%{-2553409437696000,[0,4,16,24,24,44,10,22]%%%}+%%%{-1291226242351104,[0,4,16,22,24,46

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224,[0,4,16,12,24,56,10,16]%%%}+%%%{-109773324288,[0,4,16,10,24,58,10,15]%%%}+%%%{-5360320512,[0,4,16,8,24,60,

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2567067648*i,[0,4,14,16,24,52,11,18]%%%}+%%%{-5021789847552*i,[0,4,14,14,24,54,11,17]%%%}+%%%{-609944076288*i,

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060234470195200,[0,4,12,30,24,38,12,25]%%%}+%%%{1815411173621760,[0,4,12,28,24,40,12,24]%%%}+%%%{1309434194165

760,[0,4,12,26,24,42,12,23]%%%}+%%%{771048188411904,[0,4,12,24,24,44,12,22]%%%}+%%%{368246431678464,[0,4,12,22

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%}+%%%{9903550758912,[0,4,12,16,24,52,12,18]%%%}+%%%{1701750767616,[0,4,12,14,24,54,12,17]%%%}+%%%{20299382784

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6,44,24,22,8,29]%%%}+%%%{148394475520,[0,2,26,42,24,24,8,28]%%%}+%%%{510438408192,[0,2,26,40,24,26,8,27]%%%}+%

%%{1342911283200,[0,2,26,38,24,28,8,26]%%%}+%%%{2802319360000,[0,2,26,36,24,30,8,25]%%%}+%%%{4744596684800,[0,

2,26,34,24,32,8,24]%%%}+%%%{6611027361792,[0,2,26,32,24,34,8,23]%%%}+%%%{7646216716288,[0,2,26,30,24,36,8,22]%

%%}+%%%{7373563887616,[0,2,26,28,24,38,8,21]%%%}+%%%{5936130293760,[0,2,26,26,24,40,8,20]%%%}+%%%{398313914368

0,[0,2,26,24,24,42,8,19]%%%}+%%%{2217692626944,[0,2,26,22,24,44,8,18]%%%}+%%%{1016683560960,[0,2,26,20,24,46,8

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80,[0,2,26,14,24,52,8,14]%%%}+%%%{4597481472,[0,2,26,12,24,54,8,13]%%%}+%%%{565182464,[0,2,26,10,24,56,8,12]%%

%}+%%%{43515904,[0,2,26,8,24,58,8,11]%%%}+%%%{1572864,[0,2,26,6,24,60,8,10]%%%}+%%%{2415919104*i,[0,2,24,48,24

,18,9,31]%%%}+%%%{37446746112*i,[0,2,24,46,24,20,9,30]%%%}+%%%{277126053888*i,[0,2,24,44,24,22,9,29]%%%}+%%%{1

303438688256*i,[0,2,24,42,24,24,9,28]%%%}+%%%{4373908291584*i,[0,2,24,40,24,26,9,27]%%%}+%%%{11138727149568*i,

[0,2,24,38,24,28,9,26]%%%}+%%%{22345436626944*i,[0,2,24,36,24,30,9,25]%%%}+%%%{36163768811520*i,[0,2,24,34,24,

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%%%}+%%%{37565871552,[0,0,16,34,24,30,16,21]%%%}+%%%{52882394832,[0,0,16,32,24,32,16,20]%%%}+%%%{57105054432,[

0,0,16,30,24,34,16,19]%%%}+%%%{47699939664,[0,0,16,28,24,36,16,18]%%%}+%%%{30839429376,[0,0,16,26,24,38,16,17]

%%%}+%%%{15329225376,[0,0,16,24,24,40,16,16]%%%}+%%%{5771580480,[0,0,16,22,24,42,16,15]%%%}+%%%{1604663136,[0,

0,16,20,24,44,16,14]%%%}+%%%{316187712,[0,0,16,18,24,46,16,13]%%%}+%%%{41255568,[0,0,16,16,24,48,16,12]%%%}+%%

%{3149280,[0,0,16,14,24,50,16,11]%%%}+%%%{104976,[0,0,16,12,24,52,16,10]%%%} / %%%{-1024,[0,2,2,12,4,8,2,10]%%

%}+%%%{-5120,[0,2,2,10,4,10,2,9]%%%}+%%%{-10496,[0,2,2,8,4,12,2,8]%%%}+%%%{-11264,[0,2,2,6,4,14,2,7]%%%}+%%%{-

6656,[0,2,2,4,4,16,2,6]%%%}+%%%{-2048,[0,2,2,2,4,18,2,5]%%%}+%%%{-256,[0,2,2,0,4,20,2,4]%%%}+%%%{-64,[0,0,8,12

,4,6,2,7]%%%}+%%%{-224,[0,0,8,10,4,8,2,6]%%%}+%%%{-312,[0,0,8,8,4,10,2,5]%%%}+%%%{-220,[0,0,8,6,4,12,2,4]%%%}+

%%%{-80,[0,0,8,4,4,14,2,3]%%%}+%%%{-12,[0,0,8,2,4,16,2,2]%%%}+%%%{-192*i,[0,0,6,12,4,6,3,7]%%%}+%%%{-672*i,[0,

0,6,10,4,8,3,6]%%%}+%%%{-912*i,[0,0,6,8,4,10,3,5]%%%}+%%%{-600*i,[0,0,6,6,4,12,3,4]%%%}+%%%{-192*i,[0,0,6,4,4,

14,3,3]%%%}+%%%{-24*i,[0,0,6,2,4,16,3,2]%%%}+%%%{144,[0,0,4,12,4,6,4,7]%%%}+%%%{504,[0,0,4,10,4,8,4,6]%%%}+%%%

{666,[0,0,4,8,4,10,4,5]%%%}+%%%{405,[0,0,4,6,4,12,4,4]%%%}+%%%{108,[0,0,4,4,4,14,4,3]%%%}+%%%{9,[0,0,4,2,4,16,

4,2]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [A] time = 1.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{2}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^(3/2)/(e*x^2+d)^2,x)

[Out]

int((a+b*arcsin(c*x))^(3/2)/(e*x^2+d)^2,x)

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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^(3/2)/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(3/2)/(e*x^2 + d)^2, x)

________________________________________________________________________________________

mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}}{{\left (e\,x^2+d\right )}^2} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))^(3/2)/(d + e*x^2)^2,x)

[Out]

int((a + b*asin(c*x))^(3/2)/(d + e*x^2)^2, x)

________________________________________________________________________________________

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((a+b*asin(c*x))**(3/2)/(e*x**2+d)**2,x)

[Out]

Timed out

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