3.216 \(\int \frac {(d+e x^2)^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)
Optimal. Leaf size=64 \[ \frac {x}{c}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{3/2} \sqrt {e} \sqrt {c d-b e}} \]
[Out]
x/c-(-b*e+2*c*d)*arctanh(x*c^(1/2)*e^(1/2)/(-b*e+c*d)^(1/2))/c^(3/2)/e^(1/2)/(-b*e+c*d)^(1/2)
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Rubi [A] time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{1149, 388, 208} \[ \frac {x}{c}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{3/2} \sqrt {e} \sqrt {c d-b e}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x^2)^2/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
x/c - ((2*c*d - b*e)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(3/2)*Sqrt[e]*Sqrt[c*d - b*e])
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 388
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Rule 1149
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac {d+e x^2}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\frac {x}{c}-\frac {\left (-c d e+\frac {e \left (-c d^2+b d e\right )}{d}\right ) \int \frac {1}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx}{c e}\\ &=\frac {x}{c}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{3/2} \sqrt {e} \sqrt {c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 63, normalized size = 0.98 \[ \frac {x}{c}-\frac {(b e-2 c d) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{c^{3/2} \sqrt {e} \sqrt {b e-c d}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^2)^2/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
x/c - ((-2*c*d + b*e)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(3/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])
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fricas [A] time = 1.01, size = 210, normalized size = 3.28 \[ \left [-\frac {\sqrt {c^{2} d e - b c e^{2}} {\left (2 \, c d - b e\right )} \log \left (\frac {c e x^{2} + c d - b e + 2 \, \sqrt {c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right ) - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x}{2 \, {\left (c^{3} d e - b c^{2} e^{2}\right )}}, -\frac {\sqrt {-c^{2} d e + b c e^{2}} {\left (2 \, c d - b e\right )} \arctan \left (-\frac {\sqrt {-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) - {\left (c^{2} d e - b c e^{2}\right )} x}{c^{3} d e - b c^{2} e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")
[Out]
[-1/2*(sqrt(c^2*d*e - b*c*e^2)*(2*c*d - b*e)*log((c*e*x^2 + c*d - b*e + 2*sqrt(c^2*d*e - b*c*e^2)*x)/(c*e*x^2
- c*d + b*e)) - 2*(c^2*d*e - b*c*e^2)*x)/(c^3*d*e - b*c^2*e^2), -(sqrt(-c^2*d*e + b*c*e^2)*(2*c*d - b*e)*arcta
n(-sqrt(-c^2*d*e + b*c*e^2)*x/(c*d - b*e)) - (c^2*d*e - b*c*e^2)*x)/(c^3*d*e - b*c^2*e^2)]
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giac [B] time = 4.82, size = 7051, normalized size = 110.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")
[Out]
x/c - 1/8*(32*b*c^8*d^4*e^8 - 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^4 - 64*b^2*c^7*d^3*e^9 + 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^3*e^5 + 48*
b^3*c^6*d^2*e^10 - 24*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^6 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c
*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^2*e^6 - 4*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^2*e^6 - 16*b^4*c
^5*d*e^11 - 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^6*d^2*e^6 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e
^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*e^7 - 8*sqrt(2)*sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*
c^4*d*e^7 + 4*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*
e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d*e^7 + 2*b^5*c^4*e^12 + 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^5*d*e^7
- sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e
^4)*c*e^2)*b^5*c^2*e^8 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*e^8 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*
e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*e^8 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^
4)*b^3*c^4*e^8 + (64*c^7*d^5*e^7 - 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(
4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^5*d^5*e^3 - 160*b*c^6*d^4*e^8 + 80*sqrt(2)*sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^4*d^4*e^4 + 160
*b^2*c^5*d^3*e^9 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^3*d^3*e^5 + 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*
c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^4*d^3*e^5 - 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*
c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^5*d^3*e^5 - 80*b^3*c^4*
d^2*e^10 - 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^5*d^3*e^5 + 40*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^
3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^2*d^2*e^6 - 24*sqrt(2)*sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^
2*c^3*d^2*e^6 + 12*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^4*d^2*e^6 + 20*b^4*c^3*d*e^11 + 24*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^
4*d^2*e^6 - 10*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*c*e^2)*b^4*c*d*e^7 + 12*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqr
t(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^2*d*e^7 - 6*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b
^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^3*d*e^7 - 2*b^5*c^2*e^12 - 12*
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^3*d*e^7 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sq
rt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*e^8 - 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c*e^8 + sqrt(2)*sqrt(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^
2*e^8 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^2*e^8)*c^2 - 2*(64*c^8*d^6*e^6 - 32*sqrt(2)*sqrt(b*c*e
^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^6*e^4 - 160*b*c^7*d^5*e^7 + 80*sqrt(2)*sqrt(b*c*
e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^5*e^5 + 160*b^2*c^6*d^4*e^8 - 80*sqrt(2)*sqrt
(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^4*e^6 + 16*sqrt(2)*sqrt(b*c*e^4 + sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^6 - 8*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4
*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^4*e^6 - 80*b^3*c^5*d^3*e^9 + 40*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^3*e^7 - 24*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^2*c^5*d^3*e^7 + 12*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)
*b*c^6*d^3*e^7 - 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^6*d^4*e^4 + 20*b^4*c^4*d^2*e^10 - 10*sqrt(2)*sqr
t(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^2*e^8 + 12*sqrt(2)*sqrt(b*c*e^4 + sqr
t(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^8 - 6*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^2*e^8 + 24*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^5*d^3*e^5 - 2
*b^5*c^3*d*e^11 + sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*d*e^9 - 2*
sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*e^9 + sqrt(2)*sqrt(b*c*e^4
+ sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d*e^9 - 12*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^
4)*b^2*c^4*d^2*e^6 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^3*d*e^7)*abs(c))*arctan(2*sqrt(1/2)*x*e^2
/sqrt((b*c*e^4 + sqrt(b^2*c^2*e^8 + 4*(c^2*d^2*e^2 - b*c*d*e^3)*c^2*e^4))/c^2))/((16*c^8*d^6*e^6 - 48*b*c^7*d^
5*e^7 + 56*b^2*c^6*d^4*e^8 - 8*b*c^7*d^4*e^8 + 4*c^8*d^4*e^8 - 32*b^3*c^5*d^3*e^9 + 16*b^2*c^6*d^3*e^9 - 8*b*c
^7*d^3*e^9 + 9*b^4*c^4*d^2*e^10 - 10*b^3*c^5*d^2*e^10 + 5*b^2*c^6*d^2*e^10 - b^5*c^3*d*e^11 + 2*b^4*c^4*d*e^11
- b^3*c^5*d*e^11)*c^2) + 1/8*(32*b*c^8*d^4*e^8 - 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(
b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^4 - 64*b^2*c^7*d^3*e^9 + 32*sqrt(2)*s
qrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b
^2*c^5*d^3*e^5 + 48*b^3*c^6*d^2*e^10 - 24*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - s
qrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^6 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^2*e^6 - 4*sqrt(2)*sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^
6*d^2*e^6 - 16*b^4*c^5*d*e^11 - 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^6*d^2*e^6 + 8*sqrt(2)*sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*
e^7 - 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^3*c^4*d*e^7 + 4*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d*e^7 + 2*b^5*c^4*e^12 + 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*b^2*c^5*d*e^7 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*e^8 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^
4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*e^8 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*e^8 - 2*(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*b^3*c^4*e^8 + (64*c^7*d^5*e^7 - 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*
sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^5*d^5*e^3 - 160*b*c^6*d^4*e^8 + 80*sqrt(2)
*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)
*b*c^4*d^4*e^4 + 160*b^2*c^5*d^3*e^9 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - s
qrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^3*d^3*e^5 + 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^
3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^4*d^3*e^5 - 8*sqrt(2)*sqrt(
4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^5*d
^3*e^5 - 80*b^3*c^4*d^2*e^10 - 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^5*d^3*e^5 + 40*sqrt(2)*sqrt(4*c^2*
d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^2*d^2
*e^6 - 24*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c*e^2)*b^2*c^3*d^2*e^6 + 12*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^4*d^2*e^6 + 20*b^4*c^3*d*e^11 + 24*(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*b*c^4*d^2*e^6 - 10*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c*d*e^7 + 12*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4
)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^2*d*e^7 - 6*sqrt(2)*sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^3*d*e^7 -
2*b^5*c^2*e^12 - 12*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^3*d*e^7 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*e^8 - 2*sqrt(2)*sqrt(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c*
e^8 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*c*e^2)*b^3*c^2*e^8 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^2*e^8)*c^2 - 2*(64*c^8*d^6*e^6 + 3
2*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^6*e^4 - 160*b*c^7*d^5*e^7 -
80*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^5*e^5 + 160*b^2*c^6*d^4*e
^8 + 80*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^4*e^6 - 16*sqrt(2)
*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^6 + 8*sqrt(2)*sqrt(b*c*e^4 - sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^4*e^6 - 80*b^3*c^5*d^3*e^9 - 40*sqrt(2)*sqrt(b*c*e^4 -
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^3*e^7 + 24*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*
e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^3*e^7 - 12*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e
^3 + b^2*e^4)*c*e^2)*b*c^6*d^3*e^7 - 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^6*d^4*e^4 + 20*b^4*c^4*d^2*e
^10 + 10*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^2*e^8 - 12*sqrt(2
)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^8 + 6*sqrt(2)*sqrt(b*c*e^4 -
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^2*e^8 + 24*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^
4)*b*c^5*d^3*e^5 - 2*b^5*c^3*d*e^11 - sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2
)*b^5*c^2*d*e^9 + 2*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*e^9 -
sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d*e^9 - 12*(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*b^2*c^4*d^2*e^6 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^3*d*e^7)*abs(c))*arct
an(2*sqrt(1/2)*x*e^2/sqrt((b*c*e^4 - sqrt(b^2*c^2*e^8 + 4*(c^2*d^2*e^2 - b*c*d*e^3)*c^2*e^4))/c^2))/((16*c^8*d
^6*e^6 - 48*b*c^7*d^5*e^7 + 56*b^2*c^6*d^4*e^8 - 8*b*c^7*d^4*e^8 + 4*c^8*d^4*e^8 - 32*b^3*c^5*d^3*e^9 + 16*b^2
*c^6*d^3*e^9 - 8*b*c^7*d^3*e^9 + 9*b^4*c^4*d^2*e^10 - 10*b^3*c^5*d^2*e^10 + 5*b^2*c^6*d^2*e^10 - b^5*c^3*d*e^1
1 + 2*b^4*c^4*d*e^11 - b^3*c^5*d*e^11)*c^2)
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maple [A] time = 0.00, size = 79, normalized size = 1.23 \[ -\frac {b e \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}\, c}+\frac {2 d \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}}+\frac {x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
1/c*x-1/c/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*c*e)^(1/2)*c*e*x)*b*e+2/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b
*e-c*d)*c*e)^(1/2)*c*e*x)*d
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
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mupad [B] time = 0.07, size = 52, normalized size = 0.81 \[ \frac {x}{c}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,e\,x}{\sqrt {b\,e^2-c\,d\,e}}\right )\,\left (b\,e-2\,c\,d\right )}{c^{3/2}\,\sqrt {b\,e^2-c\,d\,e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x^2)^2/(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e),x)
[Out]
x/c - (atan((c^(1/2)*e*x)/(b*e^2 - c*d*e)^(1/2))*(b*e - 2*c*d))/(c^(3/2)*(b*e^2 - c*d*e)^(1/2))
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sympy [B] time = 0.49, size = 212, normalized size = 3.31 \[ \frac {\sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- b c e \sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) + c^{2} d \sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} - \frac {\sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) \log {\left (x + \frac {b c e \sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right ) - c^{2} d \sqrt {- \frac {1}{c^{3} e \left (b e - c d\right )}} \left (b e - 2 c d\right )}{b e - 2 c d} \right )}}{2} + \frac {x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d)**2/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d)*log(x + (-b*c*e*sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d) + c**2
*d*sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d))/(b*e - 2*c*d))/2 - sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d)
*log(x + (b*c*e*sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*d) - c**2*d*sqrt(-1/(c**3*e*(b*e - c*d)))*(b*e - 2*c*
d))/(b*e - 2*c*d))/2 + x/c
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