3.873 \(\int \frac {x^{11}}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=209 \[ \frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac {b x^2 \left (b^2-7 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^4 \left (b x^2 \left (b^2-10 a c\right )+a \left (b^2-16 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^3} \]

[Out]

-1/2*b*(-7*a*c+b^2)*x^2/c^2/(-4*a*c+b^2)^2+1/4*x^8*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/4*x^4*(a*(-16*
a*c+b^2)+b*(-10*a*c+b^2)*x^2)/c/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/2*b*(30*a^2*c^2-10*a*b^2*c+b^4)*arctanh((2*c*
x^2+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/4*ln(c*x^4+b*x^2+a)/c^3

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Rubi [A]  time = 0.40, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1114, 738, 818, 773, 634, 618, 206, 628} \[ \frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac {b x^2 \left (b^2-7 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (b x^2 \left (b^2-10 a c\right )+a \left (b^2-16 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(b*(b^2 - 7*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^8*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) +
(x^4*(a*(b^2 - 16*a*c) + b*(b^2 - 10*a*c)*x^2))/(4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^
2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + Log[a + b*x^2 + c*x^
4]/(4*c^3)

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 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 1114

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

Rubi steps

\begin {align*} \int \frac {x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^5}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {x^3 (8 a+b x)}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {x \left (2 a \left (b^2-16 a c\right )+2 b \left (b^2-7 a c\right ) x\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 a b \left (b^2-7 a c\right )+\left (2 a c \left (b^2-16 a c\right )-2 b^2 \left (b^2-7 a c\right )\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {b \left (b^2-7 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^4 \left (a \left (b^2-16 a c\right )+b \left (b^2-10 a c\right ) x^2\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 244, normalized size = 1.17 \[ \frac {-\frac {2 b c \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac {2 a^3 c^2+a^2 b c \left (5 c x^2-4 b\right )+a b^3 \left (b-5 c x^2\right )+b^5 x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {32 a^3 c^3-39 a^2 b^2 c^2+50 a^2 b c^3 x^2+11 a b^4 c-30 a b^3 c^2 x^2-b^6+4 b^5 c x^2}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+c \log \left (a+b x^2+c x^4\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2 + c*x^4)^3,x]

[Out]

((-b^6 + 11*a*b^4*c - 39*a^2*b^2*c^2 + 32*a^3*c^3 + 4*b^5*c*x^2 - 30*a*b^3*c^2*x^2 + 50*a^2*b*c^3*x^2)/((b^2 -
4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*c^2 + b^5*x^2 + a*b^3*(b - 5*c*x^2) + a^2*b*c*(-4*b + 5*c*x^2))/((b^2
- 4*a*c)*(a + b*x^2 + c*x^4)^2) - (2*b*c*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*
c]])/(-b^2 + 4*a*c)^(5/2) + c*Log[a + b*x^2 + c*x^4])/(4*c^4)

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fricas [B]  time = 1.13, size = 1631, normalized size = 7.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/4*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(2*b^7*c - 23*a*b^5*c^2 + 85*a^2*b^3*c^3 - 1
00*a^3*b*c^4)*x^6 + (3*b^8 - 31*a*b^6*c + 87*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(3*a*b^7 - 34
*a^2*b^5*c + 119*a^3*b^3*c^2 - 124*a^4*b*c^3)*x^2 + ((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 1
0*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2
+ 60*a^3*b*c^3)*x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*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^6*c^2 - 12*a*b^4*c^3 + 48*a^2
*b^2*c^4 - 64*a^3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 +
48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 +
2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a))/(a^2*b^6*c^3 - 12*a^3*b^
4*c^4 + 48*a^4*b^2*c^5 - 64*a^5*c^6 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*x^8 + 2*(b^7*c^4
- 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*x^6 + (b^8*c^3 - 10*a*b^6*c^4 + 24*a^2*b^4*c^5 + 32*a^3*b^2*c^
6 - 128*a^4*c^7)*x^4 + 2*(a*b^7*c^3 - 12*a^2*b^5*c^4 + 48*a^3*b^3*c^5 - 64*a^4*b*c^6)*x^2), 1/4*(3*a^2*b^6 - 3
3*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(2*b^7*c - 23*a*b^5*c^2 + 85*a^2*b^3*c^3 - 100*a^3*b*c^4)*x^6 +
(3*b^8 - 31*a*b^6*c + 87*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(3*a*b^7 - 34*a^2*b^5*c + 119*a^
3*b^3*c^2 - 124*a^4*b*c^3)*x^2 + 2*((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30*
a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*
x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*
c)/(b^2 - 4*a*c)) + ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*
a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c +
24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*
x^2)*log(c*x^4 + b*x^2 + a))/(a^2*b^6*c^3 - 12*a^3*b^4*c^4 + 48*a^4*b^2*c^5 - 64*a^5*c^6 + (b^6*c^5 - 12*a*b^4
*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*x^8 + 2*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*x^6 + (b^
8*c^3 - 10*a*b^6*c^4 + 24*a^2*b^4*c^5 + 32*a^3*b^2*c^6 - 128*a^4*c^7)*x^4 + 2*(a*b^7*c^3 - 12*a^2*b^5*c^4 + 48
*a^3*b^3*c^5 - 64*a^4*b*c^6)*x^2)]

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giac [A]  time = 1.84, size = 306, normalized size = 1.46 \[ -\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} - 2 \, b^{5} c x^{6} + 12 \, a b^{3} c^{2} x^{6} - 4 \, a^{2} b c^{3} x^{6} - 3 \, b^{6} x^{4} + 20 \, a b^{4} c x^{4} - 22 \, a^{2} b^{2} c^{2} x^{4} + 32 \, a^{3} c^{3} x^{4} - 6 \, a b^{5} x^{2} + 40 \, a^{2} b^{3} c x^{2} - 28 \, a^{3} b c^{2} x^{2} - 3 \, a^{2} b^{4} + 18 \, a^{3} b^{2} c}{8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/2*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a
^2*c^5)*sqrt(-b^2 + 4*a*c)) - 1/8*(3*b^4*c^2*x^8 - 24*a*b^2*c^3*x^8 + 48*a^2*c^4*x^8 - 2*b^5*c*x^6 + 12*a*b^3*
c^2*x^6 - 4*a^2*b*c^3*x^6 - 3*b^6*x^4 + 20*a*b^4*c*x^4 - 22*a^2*b^2*c^2*x^4 + 32*a^3*c^3*x^4 - 6*a*b^5*x^2 + 4
0*a^2*b^3*c*x^2 - 28*a^3*b*c^2*x^2 - 3*a^2*b^4 + 18*a^3*b^2*c)/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*(c*x^4 +
b*x^2 + a)^2) + 1/4*log(c*x^4 + b*x^2 + a)/c^3

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maple [B]  time = 0.02, size = 547, normalized size = 2.62 \[ -\frac {15 a^{2} b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, c}+\frac {5 a \,b^{3} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, c^{2}}-\frac {b^{5} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, c^{3}}+\frac {4 a^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {2 a \,b^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {b^{4} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {\frac {\left (25 a^{2} c^{2}-15 a \,b^{2} c +2 b^{4}\right ) b \,x^{6}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {\left (31 a^{2} c^{2}-22 a \,b^{2} c +3 b^{4}\right ) a b \,x^{2}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {\left (32 a^{3} c^{3}+11 a^{2} b^{2} c^{2}-19 a \,b^{4} c +3 b^{6}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {3 \left (8 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right ) a^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(c*x^4+b*x^2+a)^3,x)

[Out]

1/2*(1/c^2*b*(25*a^2*c^2-15*a*b^2*c+2*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(32*a^3*c^3+11*a^2*b^2*c^2-19*a*
b^4*c+3*b^6)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+a*b*(31*a^2*c^2-22*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c
^3*x^2+3/2*a^2*(8*a^2*c^2-7*a*b^2*c+b^4)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+4/c/(16*a^2*c^2-8*a
*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*a^2-2/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*a*b^2+1/4/c^3/(16*a^2*c^2
-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*b^4-15/c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*
a*c-b^2)^(1/2))*a^2*b+5/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))
*a*b^3-1/2/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^5

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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(x^11/(c*x^4+b*x^2+a)^3,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 = 7.30, size = 2588, normalized size = 12.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^2 + c*x^4)^3,x)

[Out]

((x^4*(3*b^6 + 32*a^3*c^3 + 11*a^2*b^2*c^2 - 19*a*b^4*c))/(4*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(3*a*b
^5 - 22*a^2*b^3*c + 31*a^3*b*c^2))/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a*(a*b^4 + 8*a^3*c^2 - 7*a^2*b^
2*c))/(4*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (b*x^6*(2*b^4 + 25*a^2*c^2 - 15*a*b^2*c))/(2*c^2*(b^4 + 16*a^2*
c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) - (log((a/c^4 + ((c^3*(-(b^2*(b
^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^5))^(1/2) - 1)*((8*a)/c + (2*(c^3*(-(b^2*(b^4 + 30*a^2*c^2
- 10*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^5))^(1/2) - 1)*(2*a + b*x^2))/c + (2*b*x^2*(3*b^4 + 62*a^2*c^2 - 26*a*b^2
*c))/(c*(4*a*c - b^2)^2)))/(4*c^3) + (x^2*(b^5 + 23*a^2*b*c^2 - 9*a*b^3*c))/(c^4*(4*a*c - b^2)^2))*(a/c^4 - ((
c^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^5))^(1/2) + 1)*((8*a)/c - (2*(c^3*(-(b^2*(b^4
+ 30*a^2*c^2 - 10*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^5))^(1/2) + 1)*(2*a + b*x^2))/c + (2*b*x^2*(3*b^4 + 62*a^2*c
^2 - 26*a*b^2*c))/(c*(4*a*c - b^2)^2)))/(4*c^3) + (x^2*(b^5 + 23*a^2*b*c^2 - 9*a*b^3*c))/(c^4*(4*a*c - b^2)^2)
))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4096*a^5*
c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7)) - (b*atan(((x^2*(((b
*((6*b^5*c^3 - 52*a*b^3*c^4 + 124*a^2*b*c^5)/(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^5) + ((8*b^5*c^6 - 64*a*b^3*c^7
+ 128*a^2*b*c^8)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)
)/(2*(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^5)*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a
^3*b^4*c^6 - 5120*a^4*b^2*c^7)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(8*c^3*(4*a*c - b^2)^(5/2)) + (b*(8*b^5*c^6
- 64*a*b^3*c^7 + 128*a^2*b*c^8)*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 128
0*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(16*c^3*(4*a*c - b^2)^(5/2)*(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^
5)*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7)))/(a*(4*
a*c - b^2)^2) - (b*((b^5 + 23*a^2*b*c^2 - 9*a*b^3*c)/(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^5) + (((6*b^5*c^3 - 52*
a*b^3*c^4 + 124*a^2*b*c^5)/(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^5) + ((8*b^5*c^6 - 64*a*b^3*c^7 + 128*a^2*b*c^8)*
(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(16*a^2*c^6 +
b^4*c^4 - 8*a*b^2*c^5)*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*
a^4*b^2*c^7)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(
2*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7)) - (b^2*(
(b^5*c^6)/2 - 4*a*b^3*c^7 + 8*a^2*b*c^8)*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(c^6*(4*a*c - b^2)^5*(16*a^2*c^6 +
b^4*c^4 - 8*a*b^2*c^5))))/(2*a*(4*a*c - b^2)^(5/2))) + ((b*((8*a)/c + (8*a*c^2*(2*b^10 - 2048*a^5*c^5 + 320*a
^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 64
0*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(8*c^3*(4*a*c - b^2)^(5
/2)) + (a*b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560
*a^4*b^2*c^4 - 40*a*b^8*c))/(c*(4*a*c - b^2)^(5/2)*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*b^6*c^5
+ 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7)))/(a*(4*a*c - b^2)^2) - (b*(a/c^4 + (((8*a)/c + (8*a*c^2*(2*b^10 - 204
8*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(4096*a^5*c^8 - 4*b^10*c^3 +
80*a*b^8*c^4 - 640*a^2*b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^
2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4096*a^5*c^8 - 4*b^10*c^3 + 80*a*b^8*c^4 - 640*a^2*
b^6*c^5 + 2560*a^3*b^4*c^6 - 5120*a^4*b^2*c^7)) - (a*b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(c^4*(4*a*c - b^2)
^5)))/(2*a*(4*a*c - b^2)^(5/2)))*(32*a^2*c^6*(4*a*c - b^2)^5 + 2*b^4*c^4*(4*a*c - b^2)^5 - 16*a*b^2*c^5*(4*a*c
- b^2)^5))/(b^10 + 160*a^2*b^6*c^2 - 600*a^3*b^4*c^3 + 900*a^4*b^2*c^4 - 20*a*b^8*c))*(b^4 + 30*a^2*c^2 - 10*
a*b^2*c))/(2*c^3*(4*a*c - b^2)^(5/2))

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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(x**11/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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