3.389 \(\int \frac {x^2}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\)
Optimal. Leaf size=240 \[ \frac {\sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \]
[Out]
-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))*(b-(-4*a*c+b^2)
^(1/2))^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+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))*(b+(-4*a*c+b^2)^(1/2))^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d
-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
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Rubi [A] time = 0.30, antiderivative size = 240, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used =
{1303, 377, 205} \[ \frac {\sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \]
Antiderivative was successfully verified.
[In]
Int[x^2/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]
[Out]
-((Sqrt[b - 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^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) + (Sqrt[b + 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^2
- 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])
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 1303
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2
- 4*a*c, 0] && !IntegerQ[q] && IntegerQ[m]
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac {1-\frac {b}{\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 {1+\frac {b}{\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\\ &=\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx+\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx\\ &=\left (1-\frac {b}{\sqrt {b^2-4 a c}}\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 )+\left (1+\frac {b}{\sqrt {b^2-4 a c}}\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 )\\ &=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \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^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \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^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 227, normalized size = 0.95 \[ \frac {\frac {\sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {x \sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]
[Out]
(-((Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]
]*Sqrt[d + e*x^2])])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[b + 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[2*c*d - (b + Sqrt[b^2
- 4*a*c])*e])/Sqrt[b^2 - 4*a*c]
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fricas [B] time = 10.22, size = 3395, normalized size = 14.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="fricas")
[Out]
1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(
d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3
- 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*
c)*e^2))*log((((b^2*c - 4*a*c^2)*d^3 - (b^3 - 4*a*b*c)*d^2*e + (a*b^2 - 4*a^2*c)*d*e^2)*sqrt(d^2/((b^2*c^2 - 4
*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^
3 + (a^2*b^2 - 4*a^3*c)*e^4))*x^2 + 2*a*d^2 - (b*d^2 - 4*a*d*e)*x^2 + 2*sqrt(1/2)*((b^2 - 4*a*c)*d^2*x - ((b^3
*c - 4*a*b*c^2)*d^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e + 3*(a*b^3 - 4*a^2*b*c)*d*e^2 - 2*(a^2*b^2 - 4*a^3*c
)*e^3)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2
- 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*x)*sqrt(e*x^2 + d)*sqrt(-(b*d - 2*a*e + ((b^2*c - 4
*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*
b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))
)/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)))/x^2) - 1/4*sqrt(1/2)*sqrt(-(b*d - 2*
a*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4
- 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2
- 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log((((b^2*c - 4*a*c
^2)*d^3 - (b^3 - 4*a*b*c)*d^2*e + (a*b^2 - 4*a^2*c)*d*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*
b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))
*x^2 + 2*a*d^2 - (b*d^2 - 4*a*d*e)*x^2 - 2*sqrt(1/2)*((b^2 - 4*a*c)*d^2*x - ((b^3*c - 4*a*b*c^2)*d^3 - (b^4 -
2*a*b^2*c - 8*a^2*c^2)*d^2*e + 3*(a*b^3 - 4*a^2*b*c)*d*e^2 - 2*(a^2*b^2 - 4*a^3*c)*e^3)*sqrt(d^2/((b^2*c^2 - 4
*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^
3 + (a^2*b^2 - 4*a^3*c)*e^4))*x)*sqrt(e*x^2 + d)*sqrt(-(b*d - 2*a*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)
*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2
*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b
^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)))/x^2) - 1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e - ((b^2*c - 4*a*c^2)*d^2
- (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e
+ (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c -
4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(-(((b^2*c - 4*a*c^2)*d^3 - (b^3 - 4*a*b*c)*d^
2*e + (a*b^2 - 4*a^2*c)*d*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^
2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*x^2 - 2*a*d^2 + (b*d^2 - 4*
a*d*e)*x^2 + 2*sqrt(1/2)*((b^2 - 4*a*c)*d^2*x + ((b^3*c - 4*a*b*c^2)*d^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e
+ 3*(a*b^3 - 4*a^2*b*c)*d*e^2 - 2*(a^2*b^2 - 4*a^3*c)*e^3)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a
*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)
)*x)*sqrt(e*x^2 + d)*sqrt(-(b*d - 2*a*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2
)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*
(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 -
4*a^2*c)*e^2)))/x^2) + 1/4*sqrt(1/2)*sqrt(-(b*d - 2*a*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b
^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*
c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*
c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(-(((b^2*c - 4*a*c^2)*d^3 - (b^3 - 4*a*b*c)*d^2*e + (a*b^2 - 4*a^2*c)*d*e^
2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2
*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*x^2 - 2*a*d^2 + (b*d^2 - 4*a*d*e)*x^2 - 2*sqrt(1/2)*((b
^2 - 4*a*c)*d^2*x + ((b^3*c - 4*a*b*c^2)*d^3 - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e + 3*(a*b^3 - 4*a^2*b*c)*d*e
^2 - 2*(a^2*b^2 - 4*a^3*c)*e^3)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b
^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*x)*sqrt(e*x^2 + d)*sqrt(-(
b*d - 2*a*e - ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c
^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 +
(a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)))/x^2)
________________________________________________________________________________________
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/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),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: 0.44Unable to divide, pe
rhaps due to rounding error%%%{18446744069414584320,[4,7,8,2,3,14,2]%%%}+%%%{-2147483648,[3,8,10,8,3,10,1]%%%}
+%%%{-12884901888,[3,8,10,7,2,12,2]%%%}+%%%{463856467968,[3,8,10,6,1,14,3]%%%}+%%%{1924145348608,[3,8,10,5,0,1
6,4]%%%}+%%%{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]%%%}+%%%{104689827840
0,[3,8,8,5,4,16,2]%%%}+%%%{6210522710016,[3,8,8,4,3,18,3]%%%}+%%%{-1786706395136,[3,8,8,3,2,20,4]%%%}+%%%{-115
44872091648,[3,8,8,2,1,22,5]%%%}+%%%{4398046511104,[3,8,8,1,0,24,6]%%%}+%%%{12750684160,[3,8,7,6,7,14,0]%%%}+%
%%{-23890755584,[3,8,7,5,6,16,1]%%%}+%%%{-2103460233216,[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]%%%}+%%%{384829
0697216,[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]%%%}+%%%{1477468749824,[3,8,5,1,6,24,3]%%%}+%%%{-1099511627776,[3,8,5,0,5,26,4]%%
%}+%%%{-3221225472,[3,8,4,3,10,20,0]%%%}+%%%{57982058496,[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]%%%}+%%%{-5242880,[3,6,9,3,3,6,0]%%%}+%%%{-29360128,[3,6,9,2,2,8,1]%%%}+%%%{-335
54432,[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,[3,6,6,0,6,1
2,0]%%%}+%%%{536870912,[2,7,10,6,2,8,1]%%%}+%%%{6710886400,[2,7,10,5,1,10,2]%%%}+%%%{18253611008,[2,7,10,4,0,1
2,3]%%%}+%%%{-134217728,[2,7,9,6,4,8,0]%%%}+%%%{-5502926848,[2,7,9,5,3,10,1]%%%}+%%%{-36909875200,[2,7,9,4,2,1
2,2]%%%}+%%%{-42949672960,[2,7,9,3,1,14,3]%%%}+%%%{42949672960,[2,7,9,2,0,16,4]%%%}+%%%{956301312,[2,7,8,5,5,1
0,0]%%%}+%%%{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,[1,8,10,9,3,10,1]%%%}+%%%{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]%%%}+%%%{-268435456
00,[1,8,9,8,4,12,1]%%%}+%%%{-188978561024,[1,8,9,7,3,14,2]%%%}+%%%{146028888064,[1,8,9,6,2,16,3]%%%}+%%%{96207
2674304,[1,8,9,5,1,18,4]%%%}+%%%{-549755813888,[1,8,9,4,0,20,5]%%%}+%%%{4294967296,[1,8,8,8,6,12,0]%%%}+%%%{95
026151424,[1,8,8,7,5,14,1]%%%}+%%%{239444426752,[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,[1,8,7,3,3,22,4]%%%}+%%%{-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]%%%}+%%%{322122547
2,[1,8,4,4,10,20,0]%%%}+%%%{-12884901888,[1,8,4,3,9,22,1]%%%}+%%%{12884901888,[1,8,4,2,8,24,2]%%%}+%%%{-104857
6,[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]%%%}+%%%{943
7184,[1,6,5,0,7,14,0]%%%}+%%%{-402653184,[0,7,10,7,2,8,1]%%%}+%%%{-5637144576,[0,7,10,6,1,10,2]%%%}+%%%{-16106
127360,[0,7,10,5,0,12,3]%%%}+%%%{100663296,[0,7,9,7,4,8,0]%%%}+%%%{4160749568,[0,7,9,6,3,10,1]%%%}+%%%{3019898
8800,[0,7,9,5,2,12,2]%%%}+%%%{28991029248,[0,7,9,4,1,14,3]%%%}+%%%{-68719476736,[0,7,9,3,0,16,4]%%%}+%%%{-6878
65856,[0,7,8,6,5,10,0]%%%}+%%%{-13925089280,[0,7,8,5,4,12,1]%%%}+%%%{-48184164352,[0,7,8,4,3,14,2]%%%}+%%%{493
92123904,[0,7,8,3,2,16,3]%%%}+%%%{120259084288,[0,7,8,2,1,18,4]%%%}+%%%{-68719476736,[0,7,8,1,0,20,5]%%%}+%%%{
1845493760,[0,7,7,5,6,12,0]%%%}+%%%{19964887040,[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]%%%}+%%%{-4096,[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
________________________________________________________________________________________
maple [C] time = 0.02, size = 161, normalized size = 0.67 \[ -\frac {\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}-2 \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 ) d +d^{2}\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 )}{2 \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/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)
[Out]
-1/2*e^(1/2)*sum((_R^2-2*_R*d+d^2)/(_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))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="maxima")
[Out]
integrate(x^2/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)),x)
[Out]
int(x^2/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)
[Out]
Integral(x**2/(sqrt(d + e*x**2)*(a + b*x**2 + c*x**4)), x)
________________________________________________________________________________________