3.2.57 \(\int \frac {(d+e x^2)^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)
Optimal. Leaf size=86 \[ -\frac {(2 c d-b e)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{5/2} \sqrt {e} \sqrt {c d-b e}}+\frac {x (3 c d-b e)}{c^2}+\frac {e x^3}{3 c} \]
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Rubi [A] time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{1149, 390, 208} \begin {gather*} \frac {x (3 c d-b e)}{c^2}-\frac {(2 c d-b e)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{5/2} \sqrt {e} \sqrt {c d-b e}}+\frac {e x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
((3*c*d - b*e)*x)/c^2 + (e*x^3)/(3*c) - ((2*c*d - b*e)^2*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(5/2
)*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 390
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]
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 )^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^2}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\int \left (\frac {3 c d-b e}{c^2}+\frac {e x^2}{c}+\frac {4 c^2 d^2-4 b c d e+b^2 e^2}{c^2 \left (-c d+b e+c e x^2\right )}\right ) \, dx\\ &=\frac {(3 c d-b e) x}{c^2}+\frac {e x^3}{3 c}+\frac {(2 c d-b e)^2 \int \frac {1}{-c d+b e+c e x^2} \, dx}{c^2}\\ &=\frac {(3 c d-b e) x}{c^2}+\frac {e x^3}{3 c}-\frac {(2 c d-b e)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{5/2} \sqrt {e} \sqrt {c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 84, normalized size = 0.98 \begin {gather*} \frac {(b e-2 c d)^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{c^{5/2} \sqrt {e} \sqrt {b e-c d}}-\frac {x (b e-3 c d)}{c^2}+\frac {e x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
-(((-3*c*d + b*e)*x)/c^2) + (e*x^3)/(3*c) + ((-2*c*d + b*e)^2*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/
(c^(5/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^3}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
IntegrateAlgebraic[(d + e*x^2)^3/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4), x]
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fricas [A] time = 1.70, size = 311, normalized size = 3.62 \begin {gather*} \left [\frac {2 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {c^{2} d e - b c e^{2}} \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 ) + 6 \, {\left (3 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x}{6 \, {\left (c^{4} d e - b c^{3} e^{2}\right )}}, \frac {{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} - 3 \, {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-c^{2} d e + b c e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) + 3 \, {\left (3 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x}{3 \, {\left (c^{4} d e - b c^{3} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")
[Out]
[1/6*(2*(c^3*d*e^2 - b*c^2*e^3)*x^3 + 3*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sqrt(c^2*d*e - b*c*e^2)*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)) + 6*(3*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^
3)*x)/(c^4*d*e - b*c^3*e^2), 1/3*((c^3*d*e^2 - b*c^2*e^3)*x^3 - 3*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*sqrt(-c^2*
d*e + b*c*e^2)*arctan(-sqrt(-c^2*d*e + b*c*e^2)*x/(c*d - b*e)) + 3*(3*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*x
)/(c^4*d*e - b*c^3*e^2)]
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giac [B] time = 5.30, size = 8680, normalized size = 100.93
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")
[Out]
-1/8*(64*b*c^9*d^5*e^8 - 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*c^7*d^5*e^4 - 160*b^2*c^8*d^4*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^6*d^4*e^5 + 160*b^3
*c^7*d^3*e^10 - 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^3*c^5*d^3*e^6 + 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^2*c^6*d^3*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*c^7*d^3*e^6 - 80*b^4*c^6
*d^2*e^11 - 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^7*d^3*e^6 + 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^4*c^4*d^2*e^7 - 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^5*d^2*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^2*c^6*d^2*e^7 + 20*b^5*c^5*d*e^12 + 24*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*b^2*c^6*d^2*e^7 - 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^5*c^3*d*e^8 + 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^4*c^4*d*e^8 - 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^3*c^5*d*e^8 - 2*b^6*c^4*e
^13 - 12*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^5*d*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^6*c^2*e^9 - 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^5*c^3*e^9 +
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^4*e^9 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^4*e^9 + (128*c^8*d^6*e^7 - 64*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)*c
^6*d^6*e^3 - 384*b*c^7*d^5*e^8 + 192*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^5*d^5*e^4 + 480*b^2*c^6*d^4*e^9 - 240*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^4*d^4*e^5
+ 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*c^5*d^4*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)*c^6*d^4*e^5 - 320*b^3*c^5*d^3*e^10 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c^6*d^4*e^5 + 160*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^3*d^3*e^6 - 64*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^4*d^3*e^6 + 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*c^5*d^3*e^6 +
120*b^4*c^4*d^2*e^11 + 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^5*d^3*e^6 - 60*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^2*d^2*e^7
+ 48*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^3*d^2*e^7 - 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^4*d^2*e^7 - 24*b^5*c^3*d*e^12 - 48*(4*c^2*d^2*e^2 - 4*b*c*d*e^
3 + b^2*e^4)*b^2*c^4*d^2*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^5*c*d*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^4*c^2*d*e^8 + 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^3*d*e^8 + 2
*b^6*c^2*e^13 + 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^3*d*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^6*e^9 + 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^5*c*e
^9 - 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^2*e^9 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^2*e^9)*c^2 - 2*(128*c^9*d^7*e^6 - 6
4*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^8*d^7*e^4 - 384*b*c^8*d^6*e^7 +
192*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^7*d^6*e^5 + 480*b^2*c^7*d^5*
e^8 - 240*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^6*d^5*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)*b*c^7*d^5*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)*c^8*d^5*e^6 - 320*b^3*c^6*d^4*e^9 + 160*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^5*d^4*e^7 - 64*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^6*d^4*e^7 + 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)*b*c^7*d^4*e^7 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^7*d^5*e^4 + 120*b^4*c^5
*d^3*e^10 - 60*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^4*d^3*e^8 + 48*
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^5*d^3*e^8 - 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^6*d^3*e^8 + 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*b*c^6*d^4*e^5 - 24*b^5*c^4*d^2*e^11 + 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^5*c^3*d^2*e^9 - 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^4*c^4*d^2*e^9 + 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)*b^3*c^5*d^2*e^9
- 48*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^5*d^3*e^6 + 2*b^6*c^3*d*e^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^6*c^2*d*e^10 + 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^5*c^3*d*e^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^4*d*e^10 + 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^4*d^2*e^7 - 2*(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*b^4*c^3*d*e^8)*abs(c))*arctan(2*sqrt(1/2)*x*e^4/sqrt((b*c^3*e^8 + sqrt(b^2*c^6*e^16 + 4*(c
^4*d^2*e^6 - b*c^3*d*e^7)*c^4*e^8))/c^4))/((16*c^9*d^6*e^6 - 48*b*c^8*d^5*e^7 + 56*b^2*c^7*d^4*e^8 - 8*b*c^8*d
^4*e^8 + 4*c^9*d^4*e^8 - 32*b^3*c^6*d^3*e^9 + 16*b^2*c^7*d^3*e^9 - 8*b*c^8*d^3*e^9 + 9*b^4*c^5*d^2*e^10 - 10*b
^3*c^6*d^2*e^10 + 5*b^2*c^7*d^2*e^10 - b^5*c^4*d*e^11 + 2*b^4*c^5*d*e^11 - b^3*c^6*d*e^11)*c^2) + 1/8*(64*b*c^
9*d^5*e^8 - 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*c^7*d^5*e^4 - 160*b^2*c^8*d^4*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^6*d^4*e^5 + 160*b^3*c^7*d^3*e^10
- 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^3*c^5*d^3*e^6 + 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^2*c^6*d^3*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*c^7*d^3*e^6 - 80*b^4*c^6*d^2*e^11 - 1
6*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^7*d^3*e^6 + 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^4*c^4*d^2*e^7 - 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^5*d^2*
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^2*c^6*d^2*e^7 + 20*b^5*c^5*d*e^12 + 24*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^6*d^2*
e^7 - 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^5*c^3*d*e^8 + 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^4*c^4*d*e^8 - 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^3*c^5*d*e^8 - 2*b^6*c^4*e^13 - 12*(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^5*d*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^6*c^2*e^9 - 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^5*c^3*e^9 + 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^4*e^9 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^4*e^9 + (128*c^8*d^6*e^7 - 64*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^6*d^6*e^3 -
384*b*c^7*d^5*e^8 + 192*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^5*d^5*e^4 + 480*b^2*c^6*d^4*e^9 - 240*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^4*d^4*e^5 + 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*c^5*d^4*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)*c^6*d^4*e^5 - 320*b^3*c^5*d^3*e^10 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^
6*d^4*e^5 + 160*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^3*d^3*e^6 - 64*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^4*d^3*e^6 + 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*c^5*d^3*e^6 + 120*b^4*c^4*
d^2*e^11 + 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^5*d^3*e^6 - 60*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^2*d^2*e^7 + 48*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^3*d^2*e^7 - 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^4*d^2*e^7 - 24*b^5*c^3*d*e^12 - 48*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*
b^2*c^4*d^2*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^5*c*d*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^4*c^2*d*e^8 + 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^3*d*e^8 + 2*b^6*c^2*e^13
+ 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^3*d*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^6*e^9 + 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^5*c*e^9 - 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^2*e^9 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^2*e^9)*c^2 - 2*(128*c^9*d^7*e^6 + 64*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)*c^8*d^7*e^4 - 384*b*c^8*d^6*e^7 - 192*sqrt(2)*s
qrt(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^7*d^6*e^5 + 480*b^2*c^7*d^5*e^8 + 240*sqr
t(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^6*d^5*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)*b*c^7*d^5*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)*c^8*d^5*e^6 - 320*b^3*c^6*d^4*e^9 - 160*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^5*d^4*e^7 + 64*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^6*d^4*e^7 - 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)*b*c^7*d^4*e^7 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^7*d^5*e^4 + 120*b^4*c^5*d^3*e^10 + 6
0*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^4*d^3*e^8 - 48*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^5*d^3*e^8 + 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^6*d^3*e^8 + 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c
^6*d^4*e^5 - 24*b^5*c^4*d^2*e^11 - 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^5*c^3*d^2*e^9 + 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^4*c^4*d^2*e
^9 - 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)*b^3*c^5*d^2*e^9 - 48*(4*c^2*d
^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c^5*d^3*e^6 + 2*b^6*c^3*d*e^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^6*c^2*d*e^10 - 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^5*c^3*d*e^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^4*d*e^10 + 16*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^4*d^2*e^7 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*b^4*c^3*d*e^8)*abs(c))*arctan(2*sqrt(1/2)*x*e^4/sqrt((b*c^3*e^8 - sqrt(b^2*c^6*e^16 + 4*(c^4*d^2*e^6 -
b*c^3*d*e^7)*c^4*e^8))/c^4))/((16*c^9*d^6*e^6 - 48*b*c^8*d^5*e^7 + 56*b^2*c^7*d^4*e^8 - 8*b*c^8*d^4*e^8 + 4*c^
9*d^4*e^8 - 32*b^3*c^6*d^3*e^9 + 16*b^2*c^7*d^3*e^9 - 8*b*c^8*d^3*e^9 + 9*b^4*c^5*d^2*e^10 - 10*b^3*c^6*d^2*e^
10 + 5*b^2*c^7*d^2*e^10 - b^5*c^4*d*e^11 + 2*b^4*c^5*d*e^11 - b^3*c^6*d*e^11)*c^2) + 1/3*(c^2*x^3*e^7 + 9*c^2*
d*x*e^6 - 3*b*c*x*e^7)*e^(-6)/c^3
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maple [A] time = 0.00, size = 142, normalized size = 1.65 \begin {gather*} \frac {b^{2} e^{2} \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^{2}}-\frac {4 b d 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 {e \,x^{3}}{3 c}+\frac {4 d^{2} \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 {b e x}{c^{2}}+\frac {3 d x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
1/3*e*x^3/c-1/c^2*b*e*x+3/c*d*x+1/c^2/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*c*e)^(1/2)*c*e*x)*b^2*e^2-4/c/
((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*c*e)^(1/2)*c*e*x)*b*d*e+4/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*
c*e)^(1/2)*c*e*x)*d^2
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)^3/(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 = 4.52, size = 113, normalized size = 1.31 \begin {gather*} x\,\left (\frac {2\,d}{c}-\frac {b\,e-c\,d}{c^2}\right )+\frac {e\,x^3}{3\,c}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,e\,x\,{\left (b\,e-2\,c\,d\right )}^2}{\sqrt {b\,e^2-c\,d\,e}\,\left (b^2\,e^2-4\,b\,c\,d\,e+4\,c^2\,d^2\right )}\right )\,{\left (b\,e-2\,c\,d\right )}^2}{c^{5/2}\,\sqrt {b\,e^2-c\,d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x^2)^3/(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e),x)
[Out]
x*((2*d)/c - (b*e - c*d)/c^2) + (e*x^3)/(3*c) + (atan((c^(1/2)*e*x*(b*e - 2*c*d)^2)/((b*e^2 - c*d*e)^(1/2)*(b^
2*e^2 + 4*c^2*d^2 - 4*b*c*d*e)))*(b*e - 2*c*d)^2)/(c^(5/2)*(b*e^2 - c*d*e)^(1/2))
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sympy [B] time = 0.72, size = 275, normalized size = 3.20 \begin {gather*} x \left (- \frac {b e}{c^{2}} + \frac {3 d}{c}\right ) - \frac {\sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log {\left (x + \frac {- b c^{2} e \sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} + c^{3} d \sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac {\sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log {\left (x + \frac {b c^{2} e \sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} - c^{3} d \sqrt {- \frac {1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac {e x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d)**3/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
x*(-b*e/c**2 + 3*d/c) - sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2*log(x + (-b*c**2*e*sqrt(-1/(c**5*e*(b*e
- c*d)))*(b*e - 2*c*d)**2 + c**3*d*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2)/(b**2*e**2 - 4*b*c*d*e + 4
*c**2*d**2))/2 + sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2*log(x + (b*c**2*e*sqrt(-1/(c**5*e*(b*e - c*d))
)*(b*e - 2*c*d)**2 - c**3*d*sqrt(-1/(c**5*e*(b*e - c*d)))*(b*e - 2*c*d)**2)/(b**2*e**2 - 4*b*c*d*e + 4*c**2*d*
*2))/2 + e*x**3/(3*c)
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