3.318 \(\int \frac {1}{x^5 (a+b x^4+c x^8)} \, dx\)
Optimal. Leaf size=89 \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b x^4+c x^8\right )}{8 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{4 a x^4} \]
[Out]
-1/4/a/x^4-b*ln(x)/a^2+1/8*b*ln(c*x^8+b*x^4+a)/a^2-1/4*(-2*a*c+b^2)*arctanh((2*c*x^4+b)/(-4*a*c+b^2)^(1/2))/a^
2/(-4*a*c+b^2)^(1/2)
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Rubi [A] time = 0.12, antiderivative size = 89, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used =
{1357, 709, 800, 634, 618, 206, 628} \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b x^4+c x^8\right )}{8 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^5*(a + b*x^4 + c*x^8)),x]
[Out]
-1/(4*a*x^4) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(4*a^2*Sqrt[b^2 - 4*a*c]) - (b*Log[x])
/a^2 + (b*Log[a + b*x^4 + c*x^8])/(8*a^2)
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 618
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 628
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 634
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 709
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && LtQ[m, -1]
Rule 800
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
Rule 1357
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^4+c x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 a x^4}+\frac {\operatorname {Subst}\left (\int \frac {-b-c x}{x \left (a+b x+c x^2\right )} \, dx,x,x^4\right )}{4 a}\\ &=-\frac {1}{4 a x^4}+\frac {\operatorname {Subst}\left (\int \left (-\frac {b}{a x}+\frac {b^2-a c+b c x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^4\right )}{4 a}\\ &=-\frac {1}{4 a x^4}-\frac {b \log (x)}{a^2}+\frac {\operatorname {Subst}\left (\int \frac {b^2-a c+b c x}{a+b x+c x^2} \, dx,x,x^4\right )}{4 a^2}\\ &=-\frac {1}{4 a x^4}-\frac {b \log (x)}{a^2}+\frac {b \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^4\right )}{8 a^2}+\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^4\right )}{8 a^2}\\ &=-\frac {1}{4 a x^4}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^4+c x^8\right )}{8 a^2}-\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )}{4 a^2}\\ &=-\frac {1}{4 a x^4}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 a^2 \sqrt {b^2-4 a c}}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^4+c x^8\right )}{8 a^2}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 92, normalized size = 1.03 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\& ,\frac {\text {$\#$1}^4 b c \log (x-\text {$\#$1})-a c \log (x-\text {$\#$1})+b^2 \log (x-\text {$\#$1})}{2 \text {$\#$1}^4 c+b}\& \right ]}{4 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^5*(a + b*x^4 + c*x^8)),x]
[Out]
-1/4*1/(a*x^4) - (b*Log[x])/a^2 + RootSum[a + b*#1^4 + c*#1^8 & , (b^2*Log[x - #1] - a*c*Log[x - #1] + b*c*Log
[x - #1]*#1^4)/(b + 2*c*#1^4) & ]/(4*a^2)
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fricas [A] time = 1.00, size = 293, normalized size = 3.29 \[ \left [-\frac {{\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} x^{4} \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} x^{4} \log \left (c x^{8} + b x^{4} + a\right ) + 8 \, {\left (b^{3} - 4 \, a b c\right )} x^{4} \log \relax (x) + 2 \, a b^{2} - 8 \, a^{2} c}{8 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4}}, -\frac {2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} x^{4} \log \left (c x^{8} + b x^{4} + a\right ) + 8 \, {\left (b^{3} - 4 \, a b c\right )} x^{4} \log \relax (x) + 2 \, a b^{2} - 8 \, a^{2} c}{8 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(c*x^8+b*x^4+a),x, algorithm="fricas")
[Out]
[-1/8*((b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*x^4*log((2*c^2*x^8 + 2*b*c*x^4 + b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 -
4*a*c))/(c*x^8 + b*x^4 + a)) - (b^3 - 4*a*b*c)*x^4*log(c*x^8 + b*x^4 + a) + 8*(b^3 - 4*a*b*c)*x^4*log(x) + 2*
a*b^2 - 8*a^2*c)/((a^2*b^2 - 4*a^3*c)*x^4), -1/8*(2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*x^4*arctan(-(2*c*x^4 + b)
*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*x^4*log(c*x^8 + b*x^4 + a) + 8*(b^3 - 4*a*b*c)*x^4*log(x)
+ 2*a*b^2 - 8*a^2*c)/((a^2*b^2 - 4*a^3*c)*x^4)]
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giac [A] time = 14.50, size = 94, normalized size = 1.06 \[ \frac {b \log \left (c x^{8} + b x^{4} + a\right )}{8 \, a^{2}} - \frac {b \log \left (x^{4}\right )}{4 \, a^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} + \frac {b x^{4} - a}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(c*x^8+b*x^4+a),x, algorithm="giac")
[Out]
1/8*b*log(c*x^8 + b*x^4 + a)/a^2 - 1/4*b*log(x^4)/a^2 + 1/4*(b^2 - 2*a*c)*arctan((2*c*x^4 + b)/sqrt(-b^2 + 4*a
*c))/(sqrt(-b^2 + 4*a*c)*a^2) + 1/4*(b*x^4 - a)/(a^2*x^4)
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maple [A] time = 0.01, size = 119, normalized size = 1.34 \[ -\frac {c \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{4 \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b \ln \relax (x )}{a^{2}}+\frac {b \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{8 a^{2}}-\frac {1}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^5/(c*x^8+b*x^4+a),x)
[Out]
-1/4/x^4/a-1/a^2*b*ln(x)+1/8*b*ln(c*x^8+b*x^4+a)/a^2-1/2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1
/2))*c+1/4/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2))*b^2
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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(1/x^5/(c*x^8+b*x^4+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 details)Is 4*a*c-b^2 positive or negative?
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mupad [B] time = 2.79, size = 8817, normalized size = 99.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^5*(a + b*x^4 + c*x^8)),x)
[Out]
(atan((4*a^5*(4*a*c - b^2)^2*(5*b^7 - 23*a^3*b*c^3 + 66*a^2*b^3*c^2 - 35*a*b^5*c)*(((4*b^3 - 16*a*b*c)*(((((((
(256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2))*(2*a*c -
b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c - b^2)^(1/2)*(64*
a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (2*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2)
^2)/(a^3*(4*a*c - b^2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (b^4*c^4*(4*b^3
- 16*a*b*c)*(2*a*c - b^2)^3)/(4*a^5*(4*a*c - b^2)^(3/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))
- ((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 -
(128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b
^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a
^2*b^2)) + (((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)
/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2)
)/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (((16*a^3*b*c^7 - 96*a^2*b^3*c^6)/a^5 - ((4*b^3
- 16*a*b*c)*((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)
/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c
- 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((2*a*c - b^2)*(((
4*b^3 - 16*a*b*c)*((((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c
- 16*a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(
4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (2*b^4*c^4*(4*b^3 -
16*a*b*c)*(2*a*c - b^2)^2)/(a^3*(4*a*c - b^2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((((4*b
^3 - 16*a*b*c)*((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*
a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c
- b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + (((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^
5 + ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*
c - 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)))*(2*a*c - b^2))/(8*a
^2*(4*a*c - b^2)^(1/2))))/(8*a^2*(4*a*c - b^2)^(1/2)) + (((a^2*c^8 - 16*a*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*(
(16*a^3*b*c^7 - 96*a^2*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5 + ((4*b^3 -
16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2
)))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8
*a^2*(4*a*c - b^2)^(1/2)) + (b^4*c^4*(2*a*c - b^2)^5)/(128*a^9*(4*a*c - b^2)^(5/2))))/(c^4*(a^2*c^2 - 20*b^4 +
80*a*b^2*c)*(16*a^4*c^8 + b^8*c^4 - 8*a*b^6*c^5 + 24*a^2*b^4*c^6 - 32*a^3*b^2*c^7)) - (128*a^10*x^4*(((5*b^6
- a^3*c^3 + 26*a^2*b^2*c^2 - 25*a*b^4*c)*(c^9/a^5 + ((4*b^3 - 16*a*b*c)*((20*b*c^8)/a^4 + ((4*b^3 - 16*a*b*c)*
((72*a^3*c^8 + 124*a^2*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*((864*a^4*b*c^7 + 208*a^3*b^3*c^6)/a^5 - ((4*b^3 - 1
6*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5)
)/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c -
16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*(((2*a*c - b^2)*((((448
*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a
^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*
b^5*c^4 - 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(8*a^2*(4*a*c - b^2)^(1/2)
) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(128*a^9*(4*a*c - b^2)*(64*a^3*
c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((2*a*c - b^2)*(((4*b^3 - 16*a*b*c)*((((448*a^4*b^4*c^5 - 345
6*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2)
))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*b^5*c^4 - 4608*a^6
*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (((864*a^4*b*c
^7 + 208*a^3*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c
)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c
- b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((2*a*c - b
^2)*(((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*((((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c
)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^
(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*
(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (((864*a^4*b*c^7 + 208*a^3*b^3*c^6)/a^5 - ((4*b^3 - 1
6*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5)
)/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/
(2*(64*a^3*c - 16*a^2*b^2)) - (((72*a^3*c^8 + 124*a^2*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*((864*a^4*b*c^7 + 208
*a^3*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*
a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c
- 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(8*a^2*(4*a*c - b^2)^(1/2)) - ((((2*a*c - b^2)*((
(2*a*c - b^2)*((((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b
^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*
a*c - b^2)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(8*a^
2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(128*a^9*(
4*a*c - b^2)*(64*a^3*c - 16*a^2*b^2))))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^3*(128
0*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(1024*a^11*(4*a*c - b^2)^(3/2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*
a^2*(4*a*c - b^2)^(1/2)) - ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^4*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(8192*a^
13*(4*a*c - b^2)^2*(64*a^3*c - 16*a^2*b^2))))/(32*a^5*c^4*(a^2*c^2 - 20*b^4 + 80*a*b^2*c)) + ((5*b^7 - 23*a^3*
b*c^3 + 66*a^2*b^3*c^2 - 35*a*b^5*c)*(((4*b^3 - 16*a*b*c)*(((2*a*c - b^2)*(((2*a*c - b^2)*((((448*a^4*b^4*c^5
- 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2
*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*b^5*c^4 - 460
8*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 -
16*a*b*c)*(2*a*c - b^2)^2*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(128*a^9*(4*a*c - b^2)*(64*a^3*c - 16*a^2*b^
2))))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^3*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))
/(1024*a^11*(4*a*c - b^2)^(3/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) - ((4*b^3 - 16*a*b*c)*(
((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*((((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(12
80*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)
) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a
^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (((864*a^4*b*c^7 + 208*a^3*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b
*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*
a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(6
4*a^3*c - 16*a^2*b^2)) - (((72*a^3*c^8 + 124*a^2*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*((864*a^4*b*c^7 + 208*a^3*
b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b
^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*
a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^2*b^2)) + (((20*b*c^8)/a^4 + ((4*b
^3 - 16*a*b*c)*((72*a^3*c^8 + 124*a^2*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*((864*a^4*b*c^7 + 208*a^3*b^3*c^6)/a^
5 - ((4*b^3 - 16*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 46
08*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))
/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((((4*b^3 - 16*a*b*c)*(((2*a*c - b^
2)*((((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2
*a^5*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*
(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(8*a^2*(4*a*c -
b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(128*a^9*(4*a*c - b^2
)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((2*a*c - b^2)*(((4*b^3 - 16*a*b*c)*((((448*a^4*b^4
*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 1
6*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + ((4*b^3 - 16*a*b*c)*(2*a*c - b^2)*(1280*a^5*b^5*c^4
- 4608*a^6*b^3*c^5))/(16*a^7*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (((8
64*a^4*b*c^7 + 208*a^3*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((448*a^4*b^4*c^5 - 3456*a^5*b^2*c^6)/a^5 + ((4*b^3
- 16*a*b*c)*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(2*a^5*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2
)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(8*a^2*(4*a*c - b^2)^(1/2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b
^2)^(1/2)) - ((2*a*c - b^2)^5*(1280*a^5*b^5*c^4 - 4608*a^6*b^3*c^5))/(32768*a^15*(4*a*c - b^2)^(5/2))))/(32*a^
5*c^4*(4*a*c - b^2)^(1/2)*(a^2*c^2 - 20*b^4 + 80*a*b^2*c)))*(4*a*c - b^2)^(5/2))/(16*a^4*c^8 + b^8*c^4 - 8*a*b
^6*c^5 + 24*a^2*b^4*c^6 - 32*a^3*b^2*c^7) + (4*a^5*(4*a*c - b^2)^(5/2)*(5*b^6 - a^3*c^3 + 26*a^2*b^2*c^2 - 25*
a*b^4*c)*(((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*((((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c
^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^
3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b
^2)^(1/2)) - (2*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2)/(a^3*(4*a*c - b^2)*(64*a^3*c - 16*a^2*b^2))))/(2*(
64*a^3*c - 16*a^2*b^2)) + ((((4*b^3 - 16*a*b*c)*((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4
*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^3 - 1
6*a*b*c)*(2*a*c - b^2))/(a*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + (((256
*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4
*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c -
b^2)^(1/2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^2*b^2)) - (b*c^8)/a^5 + ((4*b^3
- 16*a*b*c)*((a^2*c^8 - 16*a*b^2*c^7)/a^5 + ((4*b^3 - 16*a*b*c)*((16*a^3*b*c^7 - 96*a^2*b^3*c^6)/a^5 - ((4*b^
3 - 16*a*b*c)*((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5 + ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^
5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3
*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + ((((4*b^3 - 16*a*b*c)*(((4*b^3
- 16*a*b*c)*((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^
2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c -
b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)) + (((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5
+ ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c
- 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2))))/(2*(64*a^3*c - 16*a^
2*b^2)) - (((16*a^3*b*c^7 - 96*a^2*b^3*c^6)/a^5 - ((4*b^3 - 16*a*b*c)*((256*a^3*b^4*c^5 - 96*a^4*b^2*c^6)/a^5
+ ((4*b^3 - 16*a*b*c)*((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c
- 16*a^2*b^2)))/(2*(64*a^3*c - 16*a^2*b^2))))/(2*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)
^(1/2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (((((((((256*a^4*b^5*c^4 - 256*a^5*b^3*c^5)/a^5 - (128*a
*b^4*c^4*(4*b^3 - 16*a*b*c))/(64*a^3*c - 16*a^2*b^2))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (16*b^4*c^4
*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/(a*(4*a*c - b^2)^(1/2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a
*c - b^2)^(1/2)) - (2*b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2)/(a^3*(4*a*c - b^2)*(64*a^3*c - 16*a^2*b^2)))
*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) - (b^4*c^4*(4*b^3 - 16*a*b*c)*(2*a*c - b^2)^3)/(4*a^5*(4*a*c - b^2
)^(3/2)*(64*a^3*c - 16*a^2*b^2)))*(2*a*c - b^2))/(8*a^2*(4*a*c - b^2)^(1/2)) + (b^4*c^4*(4*b^3 - 16*a*b*c)*(2*
a*c - b^2)^4)/(32*a^7*(4*a*c - b^2)^2*(64*a^3*c - 16*a^2*b^2))))/(c^4*(a^2*c^2 - 20*b^4 + 80*a*b^2*c)*(16*a^4*
c^8 + b^8*c^4 - 8*a*b^6*c^5 + 24*a^2*b^4*c^6 - 32*a^3*b^2*c^7)))*(2*a*c - b^2))/(4*a^2*(4*a*c - b^2)^(1/2)) -
(b*log(x))/a^2 - (log(a + b*x^4 + c*x^8)*(4*b^3 - 16*a*b*c))/(2*(64*a^3*c - 16*a^2*b^2)) - 1/(4*a*x^4)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**5/(c*x**8+b*x**4+a),x)
[Out]
Timed out
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