3.396 \(\int \frac {x^2}{(d+e x^2)^{3/2} (a+b x^2+c x^4)} \, dx\)
Optimal. Leaf size=333 \[ \frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {e x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
[Out]
-e*x/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)+c*arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-
(-4*a*c+b^2)^(1/2))^(1/2))*(d+(2*a*e-b*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/
2)))^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+c*arctan(x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(
-4*a*c+b^2)^(1/2))^(1/2))*(d+(-2*a*e+b*d)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
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Rubi [A] time = 0.66, antiderivative size = 333, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used =
{1299, 191, 1692, 377, 205} \[ \frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {e x}{\sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]
[Out]
-((e*x)/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x^2])) + (c*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c
*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]
]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (c*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*
ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + S
qrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2))
Rule 191
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 1299
Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> -Dist[
(d*e*f^2)/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(d + e*x^2)^q, x], x] + Dist[f^2/(c*d^2 - b*d*e + a*e^2),
Int[((f*x)^(m - 2)*(d + e*x^2)^(q + 1)*Simp[a*e + c*d*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 1] && LeQ[m, 3]
Rule 1692
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {x^2}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx &=\frac {\int \frac {a e+c d x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c d^2-b d e+a e^2}-\frac {(d e) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\int \left (\frac {c d+\frac {c (-b d+2 a e)}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {c d-\frac {c (-b d+2 a e)}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c d^2-b d e+a e^2}+\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac {e x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\left (c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c d^2-b d e+a e^2}+\frac {\left (c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c d^2-b d e+a e^2}\\ &=-\frac {e x}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {c \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [C] time = 6.72, size = 2119, normalized size = 6.36 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]
[Out]
((1 - b/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d
+ e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-
b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin[Sqrt[-((
(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(
((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + S
qrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] -
2*c*x^2)))]])/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[
Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b
^2 - 4*a*c] - 2*c*x^2)) + 4*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2
))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2
F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^2
*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt
[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (
-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))])/d))/(15*(b - Sqrt[b^2 - 4*a*c])*d*(-
(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(3/2)*(1 - (2*c*x^2)/(-b
+ Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] -
2*c*x^2))]) + ((1 + b/Sqrt[b^2 - 4*a*c])*x*(45*Sqrt[-(((b + Sqrt[b^2 - 4*a*c])*(-2*c*d + (b + Sqrt[b^2 - 4*a*
c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x^2*Sqrt[-(((b + Sqrt[b^2 - 4*a*c]
)*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^2*(d + e*x^2))/(d^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*Ar
cSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]] - (30*e*x^2*ArcSin
[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/d + (45*(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2
*c*x^2))]])/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)) - (30*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*x^4*ArcSin[Sqrt
[((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]])/(d^2*(b + Sqrt[b^2 - 4*a*c
] + 2*c*x^2)) + 4*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt
[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))] + (4*e*x^2*(((2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(5/2)*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/
(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, ((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x^2)/
(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])/d))/(15*(b + Sqrt[b^2 - 4*a*c])*d*(((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e
)*x^2)/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)))^(3/2)*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sq
rt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2))])
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fricas [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(x^2/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")
[Out]
Timed out
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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(x^2/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b,c]=[44,93,-37]Warning, need to choose a branch for the root of a polynomial with parameters
. This might be wrong.The choice was done assuming [a,b,c]=[-72,-7,6]Evaluation time: 0.57Unable to divide, pe
rhaps due to rounding error%%%{-2147483648,[3,8,10,8,3,10,1]%%%}+%%%{-12884901888,[3,8,10,7,2,12,2]%%%}+%%%{46
3856467968,[3,8,10,6,1,14,3]%%%}+%%%{1924145348608,[3,8,10,5,0,16,4]%%%}+%%%{536870912,[3,8,9,8,5,10,0]%%%}+%%
%{20401094656,[3,8,9,7,4,12,1]%%%}+%%%{-150323855360,[3,8,9,6,3,14,2]%%%}+%%%{-3135326126080,[3,8,9,5,2,16,3]%
%%}+%%%{-4672924418048,[3,8,9,4,1,18,4]%%%}+%%%{6047313952768,[3,8,9,3,0,20,5]%%%}+%%%{-4294967296,[3,8,8,7,6,
12,0]%%%}+%%%{-42412802048,[3,8,8,6,5,14,1]%%%}+%%%{1046898278400,[3,8,8,5,4,16,2]%%%}+%%%{6210522710016,[3,8,
8,4,3,18,3]%%%}+%%%{-1786706395136,[3,8,8,3,2,20,4]%%%}+%%%{-11544872091648,[3,8,8,2,1,22,5]%%%}+%%%{439804651
1104,[3,8,8,1,0,24,6]%%%}+%%%{12750684160,[3,8,7,6,7,14,0]%%%}+%%%{-23890755584,[3,8,7,5,6,16,1]%%%}+%%%{-2103
460233216,[3,8,7,4,5,18,2]%%%}+%%%{-3324304687104,[3,8,7,3,4,20,3]%%%}+%%%{9758165696512,[3,8,7,2,3,22,4]%%%}+
%%%{1649267441664,[3,8,7,1,2,24,5]%%%}+%%%{-4398046511104,[3,8,7,0,1,26,6]%%%}+%%%{-17985175552,[3,8,6,5,8,16,
0]%%%}+%%%{161866579968,[3,8,6,4,7,18,1]%%%}+%%%{1586990415872,[3,8,6,3,6,20,2]%%%}+%%%{-1795296329728,[3,8,6,
2,5,22,3]%%%}+%%%{-4123168604160,[3,8,6,1,4,24,4]%%%}+%%%{3848290697216,[3,8,6,0,3,26,5]%%%}+%%%{12213813248,[
3,8,5,4,9,18,0]%%%}+%%%{-171798691840,[3,8,5,3,8,20,1]%%%}+%%%{-212600881152,[3,8,5,2,7,22,2]%%%}+%%%{14774687
49824,[3,8,5,1,6,24,3]%%%}+%%%{-1099511627776,[3,8,5,0,5,26,4]%%%}+%%%{-3221225472,[3,8,4,3,10,20,0]%%%}+%%%{5
7982058496,[3,8,4,2,9,22,1]%%%}+%%%{-154618822656,[3,8,4,1,8,24,2]%%%}+%%%{103079215104,[3,8,4,0,7,26,3]%%%}+%
%%{1048576,[3,6,10,4,2,4,0]%%%}+%%%{8388608,[3,6,10,3,1,6,1]%%%}+%%%{16777216,[3,6,10,2,0,8,2]%%%}+%%%{-524288
0,[3,6,9,3,3,6,0]%%%}+%%%{-29360128,[3,6,9,2,2,8,1]%%%}+%%%{-33554432,[3,6,9,1,1,10,2]%%%}+%%%{9699328,[3,6,8,
2,4,8,0]%%%}+%%%{33554432,[3,6,8,1,3,10,1]%%%}+%%%{16777216,[3,6,8,0,2,12,2]%%%}+%%%{-7864320,[3,6,7,1,5,10,0]
%%%}+%%%{-12582912,[3,6,7,0,4,12,1]%%%}+%%%{2359296,[3,6,6,0,6,12,0]%%%}+%%%{-536870912,[2,7,10,6,2,8,1]%%%}+%
%%{18446744062703697920,[2,7,10,5,1,10,2]%%%}+%%%{-18253611008,[2,7,10,4,0,12,3]%%%}+%%%{134217728,[2,7,9,6,4,
8,0]%%%}+%%%{5502926848,[2,7,9,5,3,10,1]%%%}+%%%{36909875200,[2,7,9,4,2,12,2]%%%}+%%%{42949672960,[2,7,9,3,1,1
4,3]%%%}+%%%{-42949672960,[2,7,9,2,0,16,4]%%%}+%%%{-956301312,[2,7,8,5,5,10,0]%%%}+%%%{-18656264192,[2,7,8,4,4
,12,1]%%%}+%%%{-64961380352,[2,7,8,3,3,14,2]%%%}+%%%{8589934592,[2,7,8,2,2,16,3]%%%}+%%%{85899345920,[2,7,8,1,
1,18,4]%%%}+%%%{2642411520,[2,7,7,4,6,12,0]%%%}+%%%{27783069696,[2,7,7,3,5,14,1]%%%}+%%%{33957085184,[2,7,7,2,
4,16,2]%%%}+%%%{-73014444032,[2,7,7,1,3,18,3]%%%}+%%%{-42949672960,[2,7,7,0,2,20,4]%%%}+%%%{-3556769792,[2,7,6
,3,7,14,0]%%%}+%%%{-17716740096,[2,7,6,2,6,16,1]%%%}+%%%{12884901888,[2,7,6,1,5,18,2]%%%}+%%%{39728447488,[2,7
,6,0,4,20,3]%%%}+%%%{2340421632,[2,7,5,2,8,16,0]%%%}+%%%{2415919104,[2,7,5,1,7,18,1]%%%}+%%%{-12079595520,[2,7
,5,0,6,20,2]%%%}+%%%{-603979776,[2,7,4,1,9,18,0]%%%}+%%%{1207959552,[2,7,4,0,8,20,1]%%%}+%%%{2147483648,[1,8,1
0,9,3,10,1]%%%}+%%%{38654705664,[1,8,10,8,2,12,2]%%%}+%%%{51539607552,[1,8,10,7,1,14,3]%%%}+%%%{-274877906944,
[1,8,10,6,0,16,4]%%%}+%%%{-536870912,[1,8,9,9,5,10,0]%%%}+%%%{-26843545600,[1,8,9,8,4,12,1]%%%}+%%%{-188978561
024,[1,8,9,7,3,14,2]%%%}+%%%{146028888064,[1,8,9,6,2,16,3]%%%}+%%%{962072674304,[1,8,9,5,1,18,4]%%%}+%%%{-5497
55813888,[1,8,9,4,0,20,5]%%%}+%%%{4294967296,[1,8,8,8,6,12,0]%%%}+%%%{95026151424,[1,8,8,7,5,14,1]%%%}+%%%{239
444426752,[1,8,8,6,4,16,2]%%%}+%%%{-858993459200,[1,8,8,5,3,18,3]%%%}+%%%{-618475290624,[1,8,8,4,2,20,4]%%%}+%
%%{1099511627776,[1,8,8,3,1,22,5]%%%}+%%%{-12750684160,[1,8,7,7,7,14,0]%%%}+%%%{-136633647104,[1,8,7,6,6,16,1]
%%%}+%%%{62277025792,[1,8,7,5,5,18,2]%%%}+%%%{936302870528,[1,8,7,4,4,20,3]%%%}+%%%{-549755813888,[1,8,7,3,3,2
2,4]%%%}+%%%{-549755813888,[1,8,7,2,2,24,5]%%%}+%%%{17985175552,[1,8,6,6,8,16,0]%%%}+%%%{71940702208,[1,8,6,5,
7,18,1]%%%}+%%%{-267361714176,[1,8,6,4,6,20,2]%%%}+%%%{-137438953472,[1,8,6,3,5,22,3]%%%}+%%%{481036337152,[1,
8,6,2,4,24,4]%%%}+%%%{-12213813248,[1,8,5,5,9,18,0]%%%}+%%%{7247757312,[1,8,5,4,8,20,1]%%%}+%%%{103079215104,[
1,8,5,3,7,22,2]%%%}+%%%{-137438953472,[1,8,5,2,6,24,3]%%%}+%%%{3221225472,[1,8,4,4,10,20,0]%%%}+%%%{-128849018
88,[1,8,4,3,9,22,1]%%%}+%%%{12884901888,[1,8,4,2,8,24,2]%%%}+%%%{-1048576,[1,6,10,5,2,4,0]%%%}+%%%{-8388608,[1
,6,10,4,1,6,1]%%%}+%%%{-16777216,[1,6,10,3,0,8,2]%%%}+%%%{8388608,[1,6,9,4,3,6,0]%%%}+%%%{62914560,[1,6,9,3,2,
8,1]%%%}+%%%{150994944,[1,6,9,2,1,10,2]%%%}+%%%{134217728,[1,6,9,1,0,12,3]%%%}+%%%{-26476544,[1,6,8,3,4,8,0]%%
%}+%%%{-163577856,[1,6,8,2,3,10,1]%%%}+%%%{-301989888,[1,6,8,1,2,12,2]%%%}+%%%{-134217728,[1,6,8,0,1,14,3]%%%}
+%%%{41156608,[1,6,7,2,5,10,0]%%%}+%%%{178257920,[1,6,7,1,4,12,1]%%%}+%%%{167772160,[1,6,7,0,3,14,2]%%%}+%%%{-
31457280,[1,6,6,1,6,12,0]%%%}+%%%{-69206016,[1,6,6,0,5,14,1]%%%}+%%%{9437184,[1,6,5,0,7,14,0]%%%}+%%%{40265318
4,[0,7,10,7,2,8,1]%%%}+%%%{5637144576,[0,7,10,6,1,10,2]%%%}+%%%{16106127360,[0,7,10,5,0,12,3]%%%}+%%%{-1006632
96,[0,7,9,7,4,8,0]%%%}+%%%{-4160749568,[0,7,9,6,3,10,1]%%%}+%%%{-30198988800,[0,7,9,5,2,12,2]%%%}+%%%{-2899102
9248,[0,7,9,4,1,14,3]%%%}+%%%{68719476736,[0,7,9,3,0,16,4]%%%}+%%%{687865856,[0,7,8,6,5,10,0]%%%}+%%%{13925089
280,[0,7,8,5,4,12,1]%%%}+%%%{48184164352,[0,7,8,4,3,14,2]%%%}+%%%{-49392123904,[0,7,8,3,2,16,3]%%%}+%%%{-12025
9084288,[0,7,8,2,1,18,4]%%%}+%%%{68719476736,[0,7,8,1,0,20,5]%%%}+%%%{-1845493760,[0,7,7,5,6,12,0]%%%}+%%%{-19
964887040,[0,7,7,4,5,14,1]%%%}+%%%{-11542724608,[0,7,7,3,4,16,2]%%%}+%%%{113816633344,[0,7,7,2,3,18,3]%%%}+%%%
{-8589934592,[0,7,7,1,2,20,4]%%%}+%%%{-68719476736,[0,7,7,0,1,22,5]%%%}+%%%{2432696320,[0,7,6,4,7,14,0]%%%}+%%
%{11207180288,[0,7,6,3,6,16,1]%%%}+%%%{-28185722880,[0,7,6,2,5,18,2]%%%}+%%%{-34359738368,[0,7,6,1,4,20,3]%%%}
+%%%{60129542144,[0,7,6,0,3,22,4]%%%}+%%%{-1577058304,[0,7,5,3,8,16,0]%%%}+%%%{201326592,[0,7,5,2,7,18,1]%%%}+
%%%{14495514624,[0,7,5,1,6,20,2]%%%}+%%%{-17179869184,[0,7,5,0,5,22,3]%%%}+%%%{402653184,[0,7,4,2,9,18,0]%%%}+
%%%{-1610612736,[0,7,4,1,8,20,1]%%%}+%%%{1610612736,[0,7,4,0,7,22,2]%%%} / %%%{1024,[0,3,4,2,1,2,0]%%%}+%%%{40
96,[0,3,4,1,0,4,1]%%%}+%%%{-2560,[0,3,3,1,2,4,0]%%%}+%%%{-4096,[0,3,3,0,1,6,1]%%%}+%%%{1536,[0,3,2,0,3,6,0]%%%
} Error: Bad Argument Value
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maple [C] time = 0.03, size = 252, normalized size = 0.76 \[ -\frac {8 d \sqrt {e}}{\left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) \left (2 e \,x^{2}-2 \sqrt {e \,x^{2}+d}\, \sqrt {e}\, x +2 d \right )}-\frac {2 \sqrt {e}\, \left (\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} c d +c \,d^{3}+2 \left (2 a \,e^{2}-c \,d^{2}\right ) \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )+\left (-\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )^{2}\right )}{\left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) \left (\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{3} c +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} b e -3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} c d +8 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) a \,e^{2}-4 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) b d e +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) c \,d^{2}+b \,d^{2} e -c \,d^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x)
[Out]
-2*e^(1/2)/(4*a*e^2-4*b*d*e+4*c*d^2)*sum((_R^2*c*d+2*(2*a*e^2-c*d^2)*_R+c*d^3)/(_R^3*c+3*_R^2*b*e-3*_R^2*c*d+8
*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(-_R+(-e^(1/2)*x+(e*x^2+d)^(1/2))^2),_R=RootOf(_Z^4*c+c*d^4+(
4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z))-8*e^(1/2)*d/(4*a*e^2-4*b*d*e+4*c*d^
2)/(2*e*x^2-2*(e*x^2+d)^(1/2)*e^(1/2)*x+2*d)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")
[Out]
integrate(x^2/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x)
[Out]
int(x^2/((d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)
[Out]
Integral(x**2/((d + e*x**2)**(3/2)*(a + b*x**2 + c*x**4)), x)
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