3.3.97 \(\int \frac {x^5}{(d+e x^2) (a+b x^2+c x^4)} \, dx\) [297]

Optimal. Leaf size=158 \[ -\frac {\left (b^2 d-2 a c d-a b e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {d^2 \log \left (d+e x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 c \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1265, 1642, 648, 632, 212, 642} \begin {gather*} -\frac {\left (-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {d^2 \log \left (d+e x^2\right )}{2 e \left (a e^2-b d e+c d^2\right )}-\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 c \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

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 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {d^2}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {-a d-(b d-a e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {d^2 \log \left (d+e x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}+\frac {\text {Subst}\left (\int \frac {-a d-(b d-a 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 {d^2 \log \left (d+e x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac {(b d-a e) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^2 d-2 a c d-a b e\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2-b d e+a e^2\right )}\\ &=\frac {d^2 \log \left (d+e x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 c \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2 d-2 a c d-a b e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\left (b^2 d-2 a c d-a b e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac {d^2 \log \left (d+e x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 c \left (c d^2-b d e+a e^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 139, normalized size = 0.88 \begin {gather*} -\frac {2 e \left (-b^2 d+2 a c d+a b e\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} \left (-2 c d^2 \log \left (d+e x^2\right )+e (b d-a e) \log \left (a+b x^2+c x^4\right )\right )}{4 c \sqrt {-b^2+4 a c} e \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [A]
time = 0.23, size = 137, normalized size = 0.87

method	result	size
default	\(-\frac {\frac {\left (-a e +b d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a d -\frac {\left (-a e +b d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right )}+\frac {d^{2} \ln \left (e \,x^{2}+d \right )}{2 e \left (a \,e^{2}-d e b +c \,d^{2}\right )}\)	\(137\)
risch	\(\text {Expression too large to display}\)	\(9388\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(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*(-a*e+b*d)/c*ln(c*x^4+b*x^2+a)+2*(a*d-1/2*(-a*e+b*d)*b/c)/(4*a*c-b^2)^(1/2)*arct
an((2*c*x^2+b)/(4*a*c-b^2)^(1/2)))+1/2*d^2*ln(e*x^2+d)/e/(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^5/(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 = 33.10, size = 421, normalized size = 2.66 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} \log \left (x^{2} e + d\right ) + {\left (a b e^{2} - {\left (b^{2} - 2 \, a c\right )} d e\right )} \sqrt {b^{2} - 4 \, a c} \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 ) - {\left ({\left (b^{3} - 4 \, a b c\right )} d e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} \log \left (x^{2} e + d\right ) + 2 \, {\left (a b e^{2} - {\left (b^{2} - 2 \, a c\right )} d e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} d e - {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [A]
time = 4.52, size = 157, normalized size = 0.99 \begin {gather*} \frac {d^{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 (b d - a e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )}} + \frac {{\left (b^{2} d - 2 \, a c d - a b e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 11.05, size = 1853, normalized size = 11.73 \begin {gather*} \frac {d^2\,\ln \left (e\,x^2+d\right )}{2\,c\,d^2\,e-2\,b\,d\,e^2+2\,a\,e^3}+\frac {\ln \left (4\,a^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+8\,c^2\,d^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+5\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{7/2}+3\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{7/2}-16\,a^3\,b^3\,c\,e^4+64\,a^4\,b\,c^2\,e^4+640\,a^3\,c^4\,d^3\,e-384\,a^4\,c^3\,d\,e^3-4\,a^2\,b^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-8\,b^2\,c^2\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-6\,b^2\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+b^4\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-256\,a^2\,c^5\,d^4\,x^2-128\,a^4\,c^3\,e^4\,x^2-16\,b^4\,c^3\,d^4\,x^2+80\,a^2\,b^3\,c^2\,d^2\,e^2+96\,a^3\,b^2\,c^2\,e^4\,x^2+640\,a^3\,c^4\,d^2\,e^2\,x^2+4\,b^3\,c\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}+4\,a\,b\,e^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+48\,a\,b^4\,c^2\,d^3\,e-16\,a\,b^5\,c\,d^2\,e^2-4\,a\,b^3\,e^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-16\,b\,c^3\,d^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-6\,b^2\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+3\,b^4\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+20\,c^2\,d^3\,e\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}-352\,a^2\,b^2\,c^3\,d^3\,e-64\,a^3\,b\,c^3\,d^2\,e^2+96\,a^3\,b^2\,c^2\,d\,e^3+128\,a\,b^2\,c^4\,d^4\,x^2-16\,a^2\,b^4\,c\,e^4\,x^2+32\,b^5\,c^2\,d^3\,e\,x^2-16\,b^6\,c\,d^2\,e^2\,x^2-4\,b\,c\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{5/2}-480\,a^2\,b^2\,c^3\,d^2\,e^2\,x^2-12\,b\,c\,d^2\,e^2\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}-240\,a\,b^3\,c^3\,d^3\,e\,x^2+448\,a^2\,b\,c^4\,d^3\,e\,x^2-192\,a^3\,b\,c^3\,d\,e^3\,x^2+12\,b^2\,c^2\,d^3\,e\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-4\,b^3\,c\,d^2\,e^2\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+144\,a\,b^4\,c^2\,d^2\,e^2\,x^2+48\,a^2\,b^3\,c^2\,d\,e^3\,x^2\right )\,\left (\frac {b^3\,d}{4}+e\,\left (a^2\,c-\frac {a\,b^2}{4}+\frac {a\,b\,\sqrt {b^2-4\,a\,c}}{4}\right )-\frac {b^2\,d\,\sqrt {b^2-4\,a\,c}}{4}+\frac {a\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}-a\,b\,c\,d\right )}{4\,a^2\,c^2\,e^2-a\,b^2\,c\,e^2-4\,a\,b\,c^2\,d\,e+4\,a\,c^3\,d^2+b^3\,c\,d\,e-b^2\,c^2\,d^2}-\frac {\ln \left (4\,a^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+8\,c^2\,d^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+5\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{7/2}+3\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{7/2}+16\,a^3\,b^3\,c\,e^4-64\,a^4\,b\,c^2\,e^4-640\,a^3\,c^4\,d^3\,e+384\,a^4\,c^3\,d\,e^3-4\,a^2\,b^2\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-8\,b^2\,c^2\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-6\,b^2\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+b^4\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+256\,a^2\,c^5\,d^4\,x^2+128\,a^4\,c^3\,e^4\,x^2+16\,b^4\,c^3\,d^4\,x^2-80\,a^2\,b^3\,c^2\,d^2\,e^2-96\,a^3\,b^2\,c^2\,e^4\,x^2-640\,a^3\,c^4\,d^2\,e^2\,x^2+4\,b^3\,c\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}+4\,a\,b\,e^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}-48\,a\,b^4\,c^2\,d^3\,e+16\,a\,b^5\,c\,d^2\,e^2-4\,a\,b^3\,e^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-16\,b\,c^3\,d^4\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-6\,b^2\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+3\,b^4\,d\,e^3\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+20\,c^2\,d^3\,e\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+352\,a^2\,b^2\,c^3\,d^3\,e+64\,a^3\,b\,c^3\,d^2\,e^2-96\,a^3\,b^2\,c^2\,d\,e^3-128\,a\,b^2\,c^4\,d^4\,x^2+16\,a^2\,b^4\,c\,e^4\,x^2-32\,b^5\,c^2\,d^3\,e\,x^2+16\,b^6\,c\,d^2\,e^2\,x^2-4\,b\,c\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{5/2}+480\,a^2\,b^2\,c^3\,d^2\,e^2\,x^2-12\,b\,c\,d^2\,e^2\,x^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+240\,a\,b^3\,c^3\,d^3\,e\,x^2-448\,a^2\,b\,c^4\,d^3\,e\,x^2+192\,a^3\,b\,c^3\,d\,e^3\,x^2+12\,b^2\,c^2\,d^3\,e\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-4\,b^3\,c\,d^2\,e^2\,x^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-144\,a\,b^4\,c^2\,d^2\,e^2\,x^2-48\,a^2\,b^3\,c^2\,d\,e^3\,x^2\right )\,\left (e\,\left (\frac {a\,b^2}{4}-a^2\,c+\frac {a\,b\,\sqrt {b^2-4\,a\,c}}{4}\right )-\frac {b^3\,d}{4}-\frac {b^2\,d\,\sqrt {b^2-4\,a\,c}}{4}+\frac {a\,c\,d\,\sqrt {b^2-4\,a\,c}}{2}+a\,b\,c\,d\right )}{4\,a^2\,c^2\,e^2-a\,b^2\,c\,e^2-4\,a\,b\,c^2\,d\,e+4\,a\,c^3\,d^2+b^3\,c\,d\,e-b^2\,c^2\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

(d^2*log(d + e*x^2))/(2*a*e^3 - 2*b*d*e^2 + 2*c*d^2*e) + (log(4*a^2*e^4*(b^2 - 4*a*c)^(5/2) + 8*c^2*d^4*(b^2 -
4*a*c)^(5/2) + 5*d^2*e^2*(b^2 - 4*a*c)^(7/2) + 3*d*e^3*x^2*(b^2 - 4*a*c)^(7/2) - 16*a^3*b^3*c*e^4 + 64*a^4*b*
c^2*e^4 + 640*a^3*c^4*d^3*e - 384*a^4*c^3*d*e^3 - 4*a^2*b^2*e^4*(b^2 - 4*a*c)^(3/2) - 8*b^2*c^2*d^4*(b^2 - 4*a
*c)^(3/2) - 6*b^2*d^2*e^2*(b^2 - 4*a*c)^(5/2) + b^4*d^2*e^2*(b^2 - 4*a*c)^(3/2) - 256*a^2*c^5*d^4*x^2 - 128*a^
4*c^3*e^4*x^2 - 16*b^4*c^3*d^4*x^2 + 80*a^2*b^3*c^2*d^2*e^2 + 96*a^3*b^2*c^2*e^4*x^2 + 640*a^3*c^4*d^2*e^2*x^2
+ 4*b^3*c*d^3*e*(b^2 - 4*a*c)^(3/2) + 4*a*b*e^4*x^2*(b^2 - 4*a*c)^(5/2) + 48*a*b^4*c^2*d^3*e - 16*a*b^5*c*d^2
*e^2 - 4*a*b^3*e^4*x^2*(b^2 - 4*a*c)^(3/2) - 16*b*c^3*d^4*x^2*(b^2 - 4*a*c)^(3/2) - 6*b^2*d*e^3*x^2*(b^2 - 4*a
*c)^(5/2) + 3*b^4*d*e^3*x^2*(b^2 - 4*a*c)^(3/2) + 20*c^2*d^3*e*x^2*(b^2 - 4*a*c)^(5/2) - 352*a^2*b^2*c^3*d^3*e
- 64*a^3*b*c^3*d^2*e^2 + 96*a^3*b^2*c^2*d*e^3 + 128*a*b^2*c^4*d^4*x^2 - 16*a^2*b^4*c*e^4*x^2 + 32*b^5*c^2*d^3
*e*x^2 - 16*b^6*c*d^2*e^2*x^2 - 4*b*c*d^3*e*(b^2 - 4*a*c)^(5/2) - 480*a^2*b^2*c^3*d^2*e^2*x^2 - 12*b*c*d^2*e^2
*x^2*(b^2 - 4*a*c)^(5/2) - 240*a*b^3*c^3*d^3*e*x^2 + 448*a^2*b*c^4*d^3*e*x^2 - 192*a^3*b*c^3*d*e^3*x^2 + 12*b^
2*c^2*d^3*e*x^2*(b^2 - 4*a*c)^(3/2) - 4*b^3*c*d^2*e^2*x^2*(b^2 - 4*a*c)^(3/2) + 144*a*b^4*c^2*d^2*e^2*x^2 + 48
*a^2*b^3*c^2*d*e^3*x^2)*((b^3*d)/4 + e*(a^2*c - (a*b^2)/4 + (a*b*(b^2 - 4*a*c)^(1/2))/4) - (b^2*d*(b^2 - 4*a*c
)^(1/2))/4 + (a*c*d*(b^2 - 4*a*c)^(1/2))/2 - a*b*c*d))/(4*a*c^3*d^2 + 4*a^2*c^2*e^2 - b^2*c^2*d^2 + b^3*c*d*e
- a*b^2*c*e^2 - 4*a*b*c^2*d*e) - (log(4*a^2*e^4*(b^2 - 4*a*c)^(5/2) + 8*c^2*d^4*(b^2 - 4*a*c)^(5/2) + 5*d^2*e^
2*(b^2 - 4*a*c)^(7/2) + 3*d*e^3*x^2*(b^2 - 4*a*c)^(7/2) + 16*a^3*b^3*c*e^4 - 64*a^4*b*c^2*e^4 - 640*a^3*c^4*d^
3*e + 384*a^4*c^3*d*e^3 - 4*a^2*b^2*e^4*(b^2 - 4*a*c)^(3/2) - 8*b^2*c^2*d^4*(b^2 - 4*a*c)^(3/2) - 6*b^2*d^2*e^
2*(b^2 - 4*a*c)^(5/2) + b^4*d^2*e^2*(b^2 - 4*a*c)^(3/2) + 256*a^2*c^5*d^4*x^2 + 128*a^4*c^3*e^4*x^2 + 16*b^4*c
^3*d^4*x^2 - 80*a^2*b^3*c^2*d^2*e^2 - 96*a^3*b^2*c^2*e^4*x^2 - 640*a^3*c^4*d^2*e^2*x^2 + 4*b^3*c*d^3*e*(b^2 -
4*a*c)^(3/2) + 4*a*b*e^4*x^2*(b^2 - 4*a*c)^(5/2) - 48*a*b^4*c^2*d^3*e + 16*a*b^5*c*d^2*e^2 - 4*a*b^3*e^4*x^2*(
b^2 - 4*a*c)^(3/2) - 16*b*c^3*d^4*x^2*(b^2 - 4*a*c)^(3/2) - 6*b^2*d*e^3*x^2*(b^2 - 4*a*c)^(5/2) + 3*b^4*d*e^3*
x^2*(b^2 - 4*a*c)^(3/2) + 20*c^2*d^3*e*x^2*(b^2 - 4*a*c)^(5/2) + 352*a^2*b^2*c^3*d^3*e + 64*a^3*b*c^3*d^2*e^2
- 96*a^3*b^2*c^2*d*e^3 - 128*a*b^2*c^4*d^4*x^2 + 16*a^2*b^4*c*e^4*x^2 - 32*b^5*c^2*d^3*e*x^2 + 16*b^6*c*d^2*e^
2*x^2 - 4*b*c*d^3*e*(b^2 - 4*a*c)^(5/2) + 480*a^2*b^2*c^3*d^2*e^2*x^2 - 12*b*c*d^2*e^2*x^2*(b^2 - 4*a*c)^(5/2)
+ 240*a*b^3*c^3*d^3*e*x^2 - 448*a^2*b*c^4*d^3*e*x^2 + 192*a^3*b*c^3*d*e^3*x^2 + 12*b^2*c^2*d^3*e*x^2*(b^2 - 4
*a*c)^(3/2) - 4*b^3*c*d^2*e^2*x^2*(b^2 - 4*a*c)^(3/2) - 144*a*b^4*c^2*d^2*e^2*x^2 - 48*a^2*b^3*c^2*d*e^3*x^2)*
(e*((a*b^2)/4 - a^2*c + (a*b*(b^2 - 4*a*c)^(1/2))/4) - (b^3*d)/4 - (b^2*d*(b^2 - 4*a*c)^(1/2))/4 + (a*c*d*(b^2
- 4*a*c)^(1/2))/2 + a*b*c*d))/(4*a*c^3*d^2 + 4*a^2*c^2*e^2 - b^2*c^2*d^2 + b^3*c*d*e - a*b^2*c*e^2 - 4*a*b*c^
2*d*e)

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