3.3.99 \(\int \frac {x}{(d+e x^2) (a+b x^2+c x^4)} \, dx\) [299]

Optimal. Leaf size=133 \[ -\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac {e \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )} \]

[Out]

1/2*e*ln(e*x^2+d)/(a*e^2-b*d*e+c*d^2)-1/4*e*ln(c*x^4+b*x^2+a)/(a*e^2-b*d*e+c*d^2)-1/2*(-b*e+2*c*d)*arctanh((2*
c*x^2+b)/(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1261, 719, 31, 648, 632, 212, 642} \begin {gather*} -\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {e \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {e \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

-1/2*((2*c*d - b*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + (e
*Log[d + e*x^2])/(2*(c*d^2 - b*d*e + a*e^2)) - (e*Log[a + b*x^2 + c*x^4])/(4*(c*d^2 - b*d*e + a*e^2))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 719

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

\begin {align*} \int \frac {x}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {c d-b e-c e x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac {e^2 \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac {e \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}+\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac {e \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}-\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac {e \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 112, normalized size = 0.84 \begin {gather*} \frac {(-4 c d+2 b e) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} e \left (-2 \log \left (d+e x^2\right )+\log \left (a+b x^2+c x^4\right )\right )}{4 \sqrt {-b^2+4 a c} \left (-c d^2+e (b d-a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

((-4*c*d + 2*b*e)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]] + Sqrt[-b^2 + 4*a*c]*e*(-2*Log[d + e*x^2] + Log[a +
b*x^2 + c*x^4]))/(4*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + e*(b*d - a*e)))

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Maple [A]
time = 0.20, size = 113, normalized size = 0.85

method	result	size
default	\(-\frac {\frac {e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2}+\frac {2 \left (\frac {e b}{2}-c d \right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {e \ln \left (e \,x^{2}+d \right )}{2 a \,e^{2}-2 d e b +2 c \,d^{2}}\)	\(113\)
risch	\(\text {Expression too large to display}\)	\(3744\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(e*x^2+d)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/2/(a*e^2-b*d*e+c*d^2)*(1/2*e*ln(c*x^4+b*x^2+a)+2*(1/2*e*b-c*d)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-
b^2)^(1/2)))+1/2*e*ln(e*x^2+d)/(a*e^2-b*d*e+c*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(x/(e*x^2+d)/(c*x^4+b*x^2+a),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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 7.89, size = 329, normalized size = 2.47 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} e \log \left (x^{2} e + d\right ) + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c d - b e\right )} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, -\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \, {\left (b^{2} - 4 \, a c\right )} e \log \left (x^{2} e + d\right ) + 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, c d - b e\right )} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} - 4 \, a b c\right )} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[-1/4*((b^2 - 4*a*c)*e*log(c*x^4 + b*x^2 + a) - 2*(b^2 - 4*a*c)*e*log(x^2*e + d) + sqrt(b^2 - 4*a*c)*(2*c*d -
b*e)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)))/((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), -1/4*((b^2 - 4*a*c)*e*log(c*x^4 + b*x^2 + a) -
2*(b^2 - 4*a*c)*e*log(x^2*e + d) + 2*sqrt(-b^2 + 4*a*c)*(2*c*d - b*e)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c
)/(b^2 - 4*a*c)))/((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)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x**2+d)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Giac [A]
time = 6.40, size = 134, normalized size = 1.01 \begin {gather*} -\frac {e \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c d^{2} - b d e + a e^{2}\right )}} + \frac {e^{2} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} + \frac {{\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/4*e*log(c*x^4 + b*x^2 + a)/(c*d^2 - b*d*e + a*e^2) + 1/2*e^2*log(abs(x^2*e + d))/(c*d^2*e - b*d*e^2 + a*e^3
) + 1/2*(2*c*d - b*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((c*d^2 - b*d*e + a*e^2)*sqrt(-b^2 + 4*a*c))

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Mupad [B]
time = 8.71, size = 2434, normalized size = 18.30 \begin {gather*} \frac {e\,\ln \left (e\,x^2+d\right )}{2\,c\,d^2-2\,b\,d\,e+2\,a\,e^2}-\frac {\ln \left (36\,a^4\,c^3\,e^5-4\,a\,b^6\,e^5-4\,b^7\,e^5\,x^2+32\,a^2\,b^4\,c\,e^5+36\,a^2\,c^5\,d^4\,e-4\,a\,c^6\,d^5\,x^2-4\,b^6\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}-73\,a^3\,b^2\,c^2\,e^5-184\,a^3\,c^4\,d^2\,e^3+b^2\,c^5\,d^5\,x^2-4\,a\,b^5\,e^5\,\sqrt {b^2-4\,a\,c}+2\,a\,c^5\,d^5\,\sqrt {b^2-4\,a\,c}+16\,a\,b^5\,c\,d\,e^4-60\,a^2\,c^4\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+18\,a^3\,c^3\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}+146\,a^2\,b^2\,c^3\,d^2\,e^3-101\,a^2\,b^3\,c^2\,e^5\,x^2+120\,a^2\,c^5\,d^3\,e^2\,x^2+19\,b^4\,c^3\,d^3\,e^2\,x^2-25\,b^5\,c^2\,d^2\,e^3\,x^2-9\,a\,b^2\,c^4\,d^4\,e+184\,a^3\,b\,c^3\,d\,e^4+36\,a\,b^5\,c\,e^5\,x^2+16\,b^6\,c\,d\,e^4\,x^2+24\,a^2\,b^3\,c\,e^5\,\sqrt {b^2-4\,a\,c}-33\,a^3\,b\,c^2\,e^5\,\sqrt {b^2-4\,a\,c}+66\,a^3\,c^3\,d\,e^4\,\sqrt {b^2-4\,a\,c}+b\,c^5\,d^5\,x^2\,\sqrt {b^2-4\,a\,c}+18\,a\,b^3\,c^3\,d^3\,e^2-25\,a\,b^4\,c^2\,d^2\,e^3-72\,a^2\,b\,c^4\,d^3\,e^2-110\,a^2\,b^3\,c^2\,d\,e^4+84\,a^3\,b\,c^3\,e^5\,x^2-132\,a^3\,c^4\,d\,e^4\,x^2-7\,b^3\,c^4\,d^4\,e\,x^2+28\,a\,b^4\,c\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}+18\,a\,c^5\,d^4\,e\,x^2\,\sqrt {b^2-4\,a\,c}+16\,b^5\,c\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}-126\,a\,b^4\,c^2\,d\,e^4\,x^2+20\,a\,b^2\,c^3\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}-25\,a\,b^3\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+90\,a^2\,b\,c^3\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-78\,a^2\,b^2\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}-7\,b^2\,c^4\,d^4\,e\,x^2\,\sqrt {b^2-4\,a\,c}-106\,a\,b^2\,c^4\,d^3\,e^2\,x^2+168\,a\,b^3\,c^3\,d^2\,e^3\,x^2-272\,a^2\,b\,c^4\,d^2\,e^3\,x^2+281\,a^2\,b^2\,c^3\,d\,e^4\,x^2-5\,a\,b\,c^4\,d^4\,e\,\sqrt {b^2-4\,a\,c}+16\,a\,b^4\,c\,d\,e^4\,\sqrt {b^2-4\,a\,c}-53\,a^2\,b^2\,c^2\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}+28\,a\,b\,c^5\,d^4\,e\,x^2-92\,a^2\,c^4\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}+19\,b^3\,c^3\,d^3\,e^2\,x^2\,\sqrt {b^2-4\,a\,c}-25\,b^4\,c^2\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}+118\,a\,b^2\,c^3\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}-66\,a\,b\,c^4\,d^3\,e^2\,x^2\,\sqrt {b^2-4\,a\,c}-94\,a\,b^3\,c^2\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}+125\,a^2\,b\,c^3\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}\right )\,\left (e\,\left (\frac {b\,\sqrt {b^2-4\,a\,c}}{4}-a\,c+\frac {b^2}{4}\right )-\frac {c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{-4\,a^2\,c\,e^2+a\,b^2\,e^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,d^2-b^3\,d\,e+b^2\,c\,d^2}+\frac {\ln \left (4\,a\,b^6\,e^5-36\,a^4\,c^3\,e^5+4\,b^7\,e^5\,x^2-32\,a^2\,b^4\,c\,e^5-36\,a^2\,c^5\,d^4\,e+4\,a\,c^6\,d^5\,x^2-4\,b^6\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}+73\,a^3\,b^2\,c^2\,e^5+184\,a^3\,c^4\,d^2\,e^3-b^2\,c^5\,d^5\,x^2-4\,a\,b^5\,e^5\,\sqrt {b^2-4\,a\,c}+2\,a\,c^5\,d^5\,\sqrt {b^2-4\,a\,c}-16\,a\,b^5\,c\,d\,e^4-60\,a^2\,c^4\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}+18\,a^3\,c^3\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}-146\,a^2\,b^2\,c^3\,d^2\,e^3+101\,a^2\,b^3\,c^2\,e^5\,x^2-120\,a^2\,c^5\,d^3\,e^2\,x^2-19\,b^4\,c^3\,d^3\,e^2\,x^2+25\,b^5\,c^2\,d^2\,e^3\,x^2+9\,a\,b^2\,c^4\,d^4\,e-184\,a^3\,b\,c^3\,d\,e^4-36\,a\,b^5\,c\,e^5\,x^2-16\,b^6\,c\,d\,e^4\,x^2+24\,a^2\,b^3\,c\,e^5\,\sqrt {b^2-4\,a\,c}-33\,a^3\,b\,c^2\,e^5\,\sqrt {b^2-4\,a\,c}+66\,a^3\,c^3\,d\,e^4\,\sqrt {b^2-4\,a\,c}+b\,c^5\,d^5\,x^2\,\sqrt {b^2-4\,a\,c}-18\,a\,b^3\,c^3\,d^3\,e^2+25\,a\,b^4\,c^2\,d^2\,e^3+72\,a^2\,b\,c^4\,d^3\,e^2+110\,a^2\,b^3\,c^2\,d\,e^4-84\,a^3\,b\,c^3\,e^5\,x^2+132\,a^3\,c^4\,d\,e^4\,x^2+7\,b^3\,c^4\,d^4\,e\,x^2+28\,a\,b^4\,c\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}+18\,a\,c^5\,d^4\,e\,x^2\,\sqrt {b^2-4\,a\,c}+16\,b^5\,c\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}+126\,a\,b^4\,c^2\,d\,e^4\,x^2+20\,a\,b^2\,c^3\,d^3\,e^2\,\sqrt {b^2-4\,a\,c}-25\,a\,b^3\,c^2\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}+90\,a^2\,b\,c^3\,d^2\,e^3\,\sqrt {b^2-4\,a\,c}-78\,a^2\,b^2\,c^2\,d\,e^4\,\sqrt {b^2-4\,a\,c}-7\,b^2\,c^4\,d^4\,e\,x^2\,\sqrt {b^2-4\,a\,c}+106\,a\,b^2\,c^4\,d^3\,e^2\,x^2-168\,a\,b^3\,c^3\,d^2\,e^3\,x^2+272\,a^2\,b\,c^4\,d^2\,e^3\,x^2-281\,a^2\,b^2\,c^3\,d\,e^4\,x^2-5\,a\,b\,c^4\,d^4\,e\,\sqrt {b^2-4\,a\,c}+16\,a\,b^4\,c\,d\,e^4\,\sqrt {b^2-4\,a\,c}-53\,a^2\,b^2\,c^2\,e^5\,x^2\,\sqrt {b^2-4\,a\,c}-28\,a\,b\,c^5\,d^4\,e\,x^2-92\,a^2\,c^4\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}+19\,b^3\,c^3\,d^3\,e^2\,x^2\,\sqrt {b^2-4\,a\,c}-25\,b^4\,c^2\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}+118\,a\,b^2\,c^3\,d^2\,e^3\,x^2\,\sqrt {b^2-4\,a\,c}-66\,a\,b\,c^4\,d^3\,e^2\,x^2\,\sqrt {b^2-4\,a\,c}-94\,a\,b^3\,c^2\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}+125\,a^2\,b\,c^3\,d\,e^4\,x^2\,\sqrt {b^2-4\,a\,c}\right )\,\left (e\,\left (a\,c+\frac {b\,\sqrt {b^2-4\,a\,c}}{4}-\frac {b^2}{4}\right )-\frac {c\,d\,\sqrt {b^2-4\,a\,c}}{2}\right )}{-4\,a^2\,c\,e^2+a\,b^2\,e^2+4\,a\,b\,c\,d\,e-4\,a\,c^2\,d^2-b^3\,d\,e+b^2\,c\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

(e*log(d + e*x^2))/(2*a*e^2 + 2*c*d^2 - 2*b*d*e) - (log(36*a^4*c^3*e^5 - 4*a*b^6*e^5 - 4*b^7*e^5*x^2 + 32*a^2*
b^4*c*e^5 + 36*a^2*c^5*d^4*e - 4*a*c^6*d^5*x^2 - 4*b^6*e^5*x^2*(b^2 - 4*a*c)^(1/2) - 73*a^3*b^2*c^2*e^5 - 184*
a^3*c^4*d^2*e^3 + b^2*c^5*d^5*x^2 - 4*a*b^5*e^5*(b^2 - 4*a*c)^(1/2) + 2*a*c^5*d^5*(b^2 - 4*a*c)^(1/2) + 16*a*b
^5*c*d*e^4 - 60*a^2*c^4*d^3*e^2*(b^2 - 4*a*c)^(1/2) + 18*a^3*c^3*e^5*x^2*(b^2 - 4*a*c)^(1/2) + 146*a^2*b^2*c^3
*d^2*e^3 - 101*a^2*b^3*c^2*e^5*x^2 + 120*a^2*c^5*d^3*e^2*x^2 + 19*b^4*c^3*d^3*e^2*x^2 - 25*b^5*c^2*d^2*e^3*x^2
- 9*a*b^2*c^4*d^4*e + 184*a^3*b*c^3*d*e^4 + 36*a*b^5*c*e^5*x^2 + 16*b^6*c*d*e^4*x^2 + 24*a^2*b^3*c*e^5*(b^2 -
4*a*c)^(1/2) - 33*a^3*b*c^2*e^5*(b^2 - 4*a*c)^(1/2) + 66*a^3*c^3*d*e^4*(b^2 - 4*a*c)^(1/2) + b*c^5*d^5*x^2*(b
^2 - 4*a*c)^(1/2) + 18*a*b^3*c^3*d^3*e^2 - 25*a*b^4*c^2*d^2*e^3 - 72*a^2*b*c^4*d^3*e^2 - 110*a^2*b^3*c^2*d*e^4
+ 84*a^3*b*c^3*e^5*x^2 - 132*a^3*c^4*d*e^4*x^2 - 7*b^3*c^4*d^4*e*x^2 + 28*a*b^4*c*e^5*x^2*(b^2 - 4*a*c)^(1/2)
+ 18*a*c^5*d^4*e*x^2*(b^2 - 4*a*c)^(1/2) + 16*b^5*c*d*e^4*x^2*(b^2 - 4*a*c)^(1/2) - 126*a*b^4*c^2*d*e^4*x^2 +
20*a*b^2*c^3*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 25*a*b^3*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) + 90*a^2*b*c^3*d^2*e^3*(b
^2 - 4*a*c)^(1/2) - 78*a^2*b^2*c^2*d*e^4*(b^2 - 4*a*c)^(1/2) - 7*b^2*c^4*d^4*e*x^2*(b^2 - 4*a*c)^(1/2) - 106*a
*b^2*c^4*d^3*e^2*x^2 + 168*a*b^3*c^3*d^2*e^3*x^2 - 272*a^2*b*c^4*d^2*e^3*x^2 + 281*a^2*b^2*c^3*d*e^4*x^2 - 5*a
*b*c^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 16*a*b^4*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 53*a^2*b^2*c^2*e^5*x^2*(b^2 - 4*a*c)
^(1/2) + 28*a*b*c^5*d^4*e*x^2 - 92*a^2*c^4*d^2*e^3*x^2*(b^2 - 4*a*c)^(1/2) + 19*b^3*c^3*d^3*e^2*x^2*(b^2 - 4*a
*c)^(1/2) - 25*b^4*c^2*d^2*e^3*x^2*(b^2 - 4*a*c)^(1/2) + 118*a*b^2*c^3*d^2*e^3*x^2*(b^2 - 4*a*c)^(1/2) - 66*a*
b*c^4*d^3*e^2*x^2*(b^2 - 4*a*c)^(1/2) - 94*a*b^3*c^2*d*e^4*x^2*(b^2 - 4*a*c)^(1/2) + 125*a^2*b*c^3*d*e^4*x^2*(
b^2 - 4*a*c)^(1/2))*(e*((b*(b^2 - 4*a*c)^(1/2))/4 - a*c + b^2/4) - (c*d*(b^2 - 4*a*c)^(1/2))/2))/(a*b^2*e^2 -
4*a*c^2*d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e) + (log(4*a*b^6*e^5 - 36*a^4*c^3*e^5 + 4*b^7*e^5
*x^2 - 32*a^2*b^4*c*e^5 - 36*a^2*c^5*d^4*e + 4*a*c^6*d^5*x^2 - 4*b^6*e^5*x^2*(b^2 - 4*a*c)^(1/2) + 73*a^3*b^2*
c^2*e^5 + 184*a^3*c^4*d^2*e^3 - b^2*c^5*d^5*x^2 - 4*a*b^5*e^5*(b^2 - 4*a*c)^(1/2) + 2*a*c^5*d^5*(b^2 - 4*a*c)^
(1/2) - 16*a*b^5*c*d*e^4 - 60*a^2*c^4*d^3*e^2*(b^2 - 4*a*c)^(1/2) + 18*a^3*c^3*e^5*x^2*(b^2 - 4*a*c)^(1/2) - 1
46*a^2*b^2*c^3*d^2*e^3 + 101*a^2*b^3*c^2*e^5*x^2 - 120*a^2*c^5*d^3*e^2*x^2 - 19*b^4*c^3*d^3*e^2*x^2 + 25*b^5*c
^2*d^2*e^3*x^2 + 9*a*b^2*c^4*d^4*e - 184*a^3*b*c^3*d*e^4 - 36*a*b^5*c*e^5*x^2 - 16*b^6*c*d*e^4*x^2 + 24*a^2*b^
3*c*e^5*(b^2 - 4*a*c)^(1/2) - 33*a^3*b*c^2*e^5*(b^2 - 4*a*c)^(1/2) + 66*a^3*c^3*d*e^4*(b^2 - 4*a*c)^(1/2) + b*
c^5*d^5*x^2*(b^2 - 4*a*c)^(1/2) - 18*a*b^3*c^3*d^3*e^2 + 25*a*b^4*c^2*d^2*e^3 + 72*a^2*b*c^4*d^3*e^2 + 110*a^2
*b^3*c^2*d*e^4 - 84*a^3*b*c^3*e^5*x^2 + 132*a^3*c^4*d*e^4*x^2 + 7*b^3*c^4*d^4*e*x^2 + 28*a*b^4*c*e^5*x^2*(b^2
- 4*a*c)^(1/2) + 18*a*c^5*d^4*e*x^2*(b^2 - 4*a*c)^(1/2) + 16*b^5*c*d*e^4*x^2*(b^2 - 4*a*c)^(1/2) + 126*a*b^4*c
^2*d*e^4*x^2 + 20*a*b^2*c^3*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 25*a*b^3*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) + 90*a^2*b*
c^3*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 78*a^2*b^2*c^2*d*e^4*(b^2 - 4*a*c)^(1/2) - 7*b^2*c^4*d^4*e*x^2*(b^2 - 4*a*c)
^(1/2) + 106*a*b^2*c^4*d^3*e^2*x^2 - 168*a*b^3*c^3*d^2*e^3*x^2 + 272*a^2*b*c^4*d^2*e^3*x^2 - 281*a^2*b^2*c^3*d
*e^4*x^2 - 5*a*b*c^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 16*a*b^4*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 53*a^2*b^2*c^2*e^5*x^2
*(b^2 - 4*a*c)^(1/2) - 28*a*b*c^5*d^4*e*x^2 - 92*a^2*c^4*d^2*e^3*x^2*(b^2 - 4*a*c)^(1/2) + 19*b^3*c^3*d^3*e^2*
x^2*(b^2 - 4*a*c)^(1/2) - 25*b^4*c^2*d^2*e^3*x^2*(b^2 - 4*a*c)^(1/2) + 118*a*b^2*c^3*d^2*e^3*x^2*(b^2 - 4*a*c)
^(1/2) - 66*a*b*c^4*d^3*e^2*x^2*(b^2 - 4*a*c)^(1/2) - 94*a*b^3*c^2*d*e^4*x^2*(b^2 - 4*a*c)^(1/2) + 125*a^2*b*c
^3*d*e^4*x^2*(b^2 - 4*a*c)^(1/2))*(e*(a*c + (b*(b^2 - 4*a*c)^(1/2))/4 - b^2/4) - (c*d*(b^2 - 4*a*c)^(1/2))/2))
/(a*b^2*e^2 - 4*a*c^2*d^2 - 4*a^2*c*e^2 + b^2*c*d^2 - b^3*d*e + 4*a*b*c*d*e)

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