3.1.61 \(\int \frac {x^7 (d+e x^2+f x^4)}{(a+b x^2+c x^4)^2} \, dx\) [61]
Optimal. Leaf size=320 \[ \frac {\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac {\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
[Out]
1/2*(2*b^2*c*e-6*a*c^2*e-3*b^3*f-b*c*(-11*a*f+c*d))*x^2/c^3/(-4*a*c+b^2)+1/4*(4*c^2*d+3*b^2*f-2*c*(4*a*f+b*e))
*x^4/c^2/(-4*a*c+b^2)+1/2*x^6*(2*a*c*e-b*(a*f+c*d)-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x^2)/c/(-4*a*c+b^2)/(c*x^4+b
*x^2+a)-1/2*(2*b^4*c*e-12*a*b^2*c^2*e+12*a^2*c^3*e-3*b^5*f-b^3*c*(-20*a*f+c*d)+6*a*b*c^2*(-5*a*f+c*d))*arctanh
((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c+b^2)^(3/2)+1/4*(c^2*d+3*b^2*f-2*c*(a*f+b*e))*ln(c*x^4+b*x^2+a)/c^
4
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Rubi [A]
time = 0.81, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1677, 1658,
814, 648, 632, 212, 642} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (12 a^2 c^3 e-b^3 c (c d-20 a f)-12 a b^2 c^2 e+6 a b c^2 (c d-5 a f)-3 b^5 f+2 b^4 c e\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {x^4 \left (-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right )}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (-\left (x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )}{4 c^4}+\frac {x^2 \left (-b c (c d-11 a f)-6 a c^2 e-3 b^3 f+2 b^2 c e\right )}{2 c^3 \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]
[Out]
((2*b^2*c*e - 6*a*c^2*e - 3*b^3*f - b*c*(c*d - 11*a*f))*x^2)/(2*c^3*(b^2 - 4*a*c)) + ((4*c^2*d + 3*b^2*f - 2*c
*(b*e + 4*a*f))*x^4)/(4*c^2*(b^2 - 4*a*c)) + (x^6*(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*
f)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*c*e - 12*a*b^2*c^2*e + 12*a^2*c^3*e - 3*b^5*f - b^3
*c*(c*d - 20*a*f) + 6*a*b*c^2*(c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(3
/2)) + ((c^2*d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^4)
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 814
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 1658
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((f*b - 2*a*g +
(2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x
+ c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c
*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && N
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] || !IntegerQ[m
] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2,
0]))
Rule 1677
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {x^7 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 \left (2 a e-\frac {b (c d+a f)}{c}\right )-\frac {\left (4 c^2 d-2 b c e+3 b^2 f-8 a c f\right ) x}{c}\right )}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \left (-\frac {2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)}{c^3}-\frac {\left (4 c^2 d-2 b c e+3 b^2 f-8 a c f\right ) x}{c^2}-\frac {-a \left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right )+\left (b^2-4 a c\right ) \left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac {\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {-a \left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right )+\left (b^2-4 a c\right ) \left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac {\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4 \left (b^2-4 a c\right )}+\frac {\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac {\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac {\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}-\frac {\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c e-6 a c^2 e-3 b^3 f-b c (c d-11 a f)\right ) x^2}{2 c^3 \left (b^2-4 a c\right )}+\frac {\left (4 c^2 d+3 b^2 f-2 c (b e+4 a f)\right ) x^4}{4 c^2 \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (2 b^4 c e-12 a b^2 c^2 e+12 a^2 c^3 e-3 b^5 f-b^3 c (c d-20 a f)+6 a b c^2 (c d-5 a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 309, normalized size = 0.97 \begin {gather*} \frac {2 c (c e-2 b f) x^2+c^2 f x^4+\frac {2 \left (2 a^3 c^2 f+b^3 \left (c^2 d-b c e+b^2 f\right ) x^2+a b \left (b^3 f-3 c^3 d x^2+b c^2 \left (d+4 e x^2\right )-b^2 c \left (e+5 f x^2\right )\right )+a^2 c \left (-4 b^2 f-2 c^2 \left (d+e x^2\right )+b c \left (3 e+5 f x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 \left (-2 b^4 c e+12 a b^2 c^2 e-12 a^2 c^3 e+3 b^5 f+b^3 c (c d-20 a f)+6 a b c^2 (-c d+5 a f)\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\left (c^2 d+3 b^2 f-2 c (b e+a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]
[Out]
(2*c*(c*e - 2*b*f)*x^2 + c^2*f*x^4 + (2*(2*a^3*c^2*f + b^3*(c^2*d - b*c*e + b^2*f)*x^2 + a*b*(b^3*f - 3*c^3*d*
x^2 + b*c^2*(d + 4*e*x^2) - b^2*c*(e + 5*f*x^2)) + a^2*c*(-4*b^2*f - 2*c^2*(d + e*x^2) + b*c*(3*e + 5*f*x^2)))
)/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(-2*b^4*c*e + 12*a*b^2*c^2*e - 12*a^2*c^3*e + 3*b^5*f + b^3*c*(c*d
- 20*a*f) + 6*a*b*c^2*(-(c*d) + 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c^2*
d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b*x^2 + c*x^4])/(4*c^4)
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Maple [A]
time = 0.15, size = 432, normalized size = 1.35
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (-c \,x^{2} f +2 b f -c e \right )^{2}}{4 c^{4} f}+\frac {\frac {-\frac {\left (5 a^{2} b \,c^{2} f -2 a^{2} c^{3} e -5 a \,b^{3} c f +4 a \,b^{2} c^{2} e -3 a b \,c^{3} d +b^{5} f -b^{4} c e +b^{3} c^{2} d \right ) x^{2}}{c \left (4 a c -b^{2}\right )}-\frac {a \left (2 a^{2} c^{2} f -4 a \,b^{2} c f +3 a b \,c^{2} e -2 c^{3} a d +b^{4} f -b^{3} c e +b^{2} c^{2} d \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (-8 a^{2} c^{2} f +14 a \,b^{2} c f -8 a b \,c^{2} e +4 c^{3} a d -3 b^{4} f +2 b^{3} c e -b^{2} c^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (11 a^{2} b c f -6 a^{2} c^{2} e -3 a \,b^{3} f +2 a \,b^{2} c e -a b \,c^{2} d -\frac {\left (-8 a^{2} c^{2} f +14 a \,b^{2} c f -8 a b \,c^{2} e +4 c^{3} a d -3 b^{4} f +2 b^{3} c e -b^{2} c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 c^{3}}\) |
\(432\) |
| risch |
\(\text {Expression too large to display}\) |
\(7053\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
[Out]
1/4*(-c*f*x^2+2*b*f-c*e)^2/c^4/f+1/2/c^3*((-(5*a^2*b*c^2*f-2*a^2*c^3*e-5*a*b^3*c*f+4*a*b^2*c^2*e-3*a*b*c^3*d+b
^5*f-b^4*c*e+b^3*c^2*d)/c/(4*a*c-b^2)*x^2-a*(2*a^2*c^2*f-4*a*b^2*c*f+3*a*b*c^2*e-2*a*c^3*d+b^4*f-b^3*c*e+b^2*c
^2*d)/c/(4*a*c-b^2))/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)*(1/2*(-8*a^2*c^2*f+14*a*b^2*c*f-8*a*b*c^2*e+4*a*c^3*d-3*b^4
*f+2*b^3*c*e-b^2*c^2*d)/c*ln(c*x^4+b*x^2+a)+2*(11*a^2*b*c*f-6*a^2*c^2*e-3*a*b^3*f+2*a*b^2*c*e-a*b*c^2*d-1/2*(-
8*a^2*c^2*f+14*a*b^2*c*f-8*a*b*c^2*e+4*a*c^3*d-3*b^4*f+2*b^3*c*e-b^2*c^2*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c
*x^2+b)/(4*a*c-b^2)^(1/2))))
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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^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^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(4*a*c-b^2>0)', see `assume?` f
or more deta
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1043 vs.
\(2 (306) = 612\).
time = 0.63, size = 2111, normalized size = 6.60 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
[Out]
[1/4*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*f*x^8 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e - 3*(b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*f)*x^6 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e - (4*b^6*c - 33*a*b^4*c^2 + 72*
a^2*b^2*c^3 - 16*a^3*c^4)*f)*x^4 + 2*((b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d - (b^6*c - 9*a*b^4*c^2 + 26*a^2
*b^2*c^3 - 24*a^3*c^4)*e + (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*f)*x^2 - (((b^3*c^3 - 6*a*b*c^4)
*d - 2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 - 6*
a*b^2*c^3)*d - 2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*f)*x^2 + (a*b^3
*c^2 - 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)
*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)) + 2*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3*c^4)*d - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*e + 2*(a*b^6
- 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*f + (((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 2*(b^5*c^2 - 8*a*b^
3*c^3 + 16*a^2*b*c^4)*e + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*f)*x^4 + ((b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*d - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32
*a^3*b*c^3)*f)*x^2 + (a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e
+ (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^4 - 8*a^2*b^2*c^
5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^2), 1/4*(
(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*f*x^8 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e - 3*(b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*f)*x^6 + (2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^
2*c^3 - 16*a^3*c^4)*f)*x^4 + 2*((b^5*c^2 - 7*a*b^3*c^3 + 12*a^2*b*c^4)*d - (b^6*c - 9*a*b^4*c^2 + 26*a^2*b^2*c
^3 - 24*a^3*c^4)*e + (b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*f)*x^2 + 2*(((b^3*c^3 - 6*a*b*c^4)*d -
2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 - 6*a*b^
2*c^3)*d - 2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*f)*x^2 + (a*b^3*c^2
- 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)*sqr
t(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + 2*(a*b^4*c^2 - 6*a^2*b^2*c^3 + 8*a^3
*c^4)*d - 2*(a*b^5*c - 7*a^2*b^3*c^2 + 12*a^3*b*c^3)*e + 2*(a*b^6 - 8*a^2*b^4*c + 18*a^3*b^2*c^2 - 8*a^4*c^3)*
f + (((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d - 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e + (3*b^6*c - 26*a*b^
4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*f)*x^4 + ((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d - 2*(b^6*c - 8*a*b^4*c
^2 + 16*a^2*b^2*c^3)*e + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*f)*x^2 + (a*b^4*c^2 - 8*a^2*b^2*
c^3 + 16*a^3*c^4)*d - 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*e + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2
- 32*a^4*c^3)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 + 16
*a^2*c^7)*x^4 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^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**7*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)
[Out]
Timed out
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Giac [A]
time = 5.53, size = 424, normalized size = 1.32 \begin {gather*} -\frac {{\left (b^{3} c^{2} d - 6 \, a b c^{3} d + 3 \, b^{5} f - 20 \, a b^{3} c f + 30 \, a^{2} b c^{2} f - 2 \, b^{4} c e + 12 \, a b^{2} c^{2} e - 12 \, a^{2} c^{3} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} c^{3} d x^{4} - 4 \, a c^{4} d x^{4} + 3 \, b^{4} c f x^{4} - 14 \, a b^{2} c^{2} f x^{4} + 8 \, a^{2} c^{3} f x^{4} - 2 \, b^{3} c^{2} x^{4} e + 8 \, a b c^{3} x^{4} e - b^{3} c^{2} d x^{2} + 2 \, a b c^{3} d x^{2} + b^{5} f x^{2} - 4 \, a b^{3} c f x^{2} - 2 \, a^{2} b c^{2} f x^{2} + 4 \, a^{2} c^{3} x^{2} e - a b^{2} c^{2} d + a b^{4} f - 6 \, a^{2} b^{2} c f + 4 \, a^{3} c^{2} f + 2 \, a^{2} b c^{2} e}{4 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} {\left (c x^{4} + b x^{2} + a\right )}} + \frac {{\left (c^{2} d + 3 \, b^{2} f - 2 \, a c f - 2 \, b c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac {c^{2} f x^{4} - 4 \, b c f x^{2} + 2 \, c^{2} x^{2} e}{4 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
[Out]
-1/2*(b^3*c^2*d - 6*a*b*c^3*d + 3*b^5*f - 20*a*b^3*c*f + 30*a^2*b*c^2*f - 2*b^4*c*e + 12*a*b^2*c^2*e - 12*a^2*
c^3*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) - 1/4*(b^2*c^3*d*x^4
- 4*a*c^4*d*x^4 + 3*b^4*c*f*x^4 - 14*a*b^2*c^2*f*x^4 + 8*a^2*c^3*f*x^4 - 2*b^3*c^2*x^4*e + 8*a*b*c^3*x^4*e - b
^3*c^2*d*x^2 + 2*a*b*c^3*d*x^2 + b^5*f*x^2 - 4*a*b^3*c*f*x^2 - 2*a^2*b*c^2*f*x^2 + 4*a^2*c^3*x^2*e - a*b^2*c^2
*d + a*b^4*f - 6*a^2*b^2*c*f + 4*a^3*c^2*f + 2*a^2*b*c^2*e)/((b^2*c^4 - 4*a*c^5)*(c*x^4 + b*x^2 + a)) + 1/4*(c
^2*d + 3*b^2*f - 2*a*c*f - 2*b*c*e)*log(c*x^4 + b*x^2 + a)/c^4 + 1/4*(c^2*f*x^4 - 4*b*c*f*x^2 + 2*c^2*x^2*e)/c
^4
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Mupad [B]
time = 1.33, size = 2500, normalized size = 7.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x)
[Out]
x^2*(e/(2*c^2) - (b*f)/c^3) - ((2*a^3*c^2*f - 2*a^2*c^3*d + a*b^4*f - a*b^3*c*e + a*b^2*c^2*d + 3*a^2*b*c^2*e
- 4*a^2*b^2*c*f)/(2*c*(4*a*c - b^2)) + (x^2*(b^5*f - 2*a^2*c^3*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d - 5*a*b^3
*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f))/(2*c*(4*a*c - b^2)))/(a*c^3 + c^4*x^4 + b*c^3*x^2) - (log(a + b*x^2 + c
*x^4)*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*
e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))
/(2*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6)) + (f*x^4)/(4*c^2) + (atan(((8*a*c^7*(4*a*c - b
^2)^3 - 2*b^2*c^6*(4*a*c - b^2)^3)*(((((16*a^2*c^5*f - 8*a*c^6*d + 16*a*b*c^5*e - 24*a*b^2*c^4*f)/c^6 - (8*a*c
^2*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e +
336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(2
56*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6))*(3*b^5*f - 12*a^2*c^3*e + b^3*c^2*d - 2*b^4*c*e - 6*
a*b*c^3*d - 20*a*b^3*c*f + 12*a*b^2*c^2*e + 30*a^2*b*c^2*f))/(8*c^4*(4*a*c - b^2)^(3/2)) - (a*(3*b^5*f - 12*a^
2*c^3*e + b^3*c^2*d - 2*b^4*c*e - 6*a*b*c^3*d - 20*a*b^3*c*f + 12*a*b^2*c^2*e + 30*a^2*b*c^2*f)*(6*b^8*f - 128
*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*
f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(c^2*(4*a*c - b^2)^
(3/2)*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6)))/(a*(4*a*c - b^2)) - x^2*(((((24*a^2*c^7*e -
6*b^3*c^6*d + 12*b^4*c^5*e - 18*b^5*c^4*f + 28*a*b*c^7*d - 56*a*b^2*c^6*e + 96*a*b^3*c^5*f - 92*a^2*b*c^6*f)/
(4*a*c^7 - b^2*c^6) - ((8*b^3*c^8 - 32*a*b*c^9)*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7
*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*
c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(2*(4*a*c^7 - b^2*c^6)*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 19
2*a^2*b^2*c^6)))*(3*b^5*f - 12*a^2*c^3*e + b^3*c^2*d - 2*b^4*c*e - 6*a*b*c^3*d - 20*a*b^3*c*f + 12*a*b^2*c^2*e
+ 30*a^2*b*c^2*f))/(8*c^4*(4*a*c - b^2)^(3/2)) - ((8*b^3*c^8 - 32*a*b*c^9)*(3*b^5*f - 12*a^2*c^3*e + b^3*c^2*
d - 2*b^4*c*e - 6*a*b*c^3*d - 20*a*b^3*c*f + 12*a*b^2*c^2*e + 30*a^2*b*c^2*f)*(6*b^8*f - 128*a^3*c^5*d + 2*b^6
*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^
3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(16*c^4*(4*a*c - b^2)^(3/2)*(4*a*c^7
- b^2*c^6)*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6)))/(a*(4*a*c - b^2)) + (b*((((24*a^2*c^7*
e - 6*b^3*c^6*d + 12*b^4*c^5*e - 18*b^5*c^4*f + 28*a*b*c^7*d - 56*a*b^2*c^6*e + 96*a*b^3*c^5*f - 92*a^2*b*c^6*
f)/(4*a*c^7 - b^2*c^6) - ((8*b^3*c^8 - 32*a*b*c^9)*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*
b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b
^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(2*(4*a*c^7 - b^2*c^6)*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 -
192*a^2*b^2*c^6)))*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 19
2*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256
*a^3*b*c^4*e))/(2*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6)) - (9*b^7*f^2 + b^3*c^4*d^2 + 4*b
^5*c^2*e^2 - 20*a*b^3*c^3*e^2 + 12*a^2*b*c^4*e^2 - 38*a^3*b*c^3*f^2 - 12*b^6*c*e*f + 91*a^2*b^3*c^2*f^2 - 5*a*
b*c^5*d^2 - 57*a*b^5*c*f^2 - 6*a^2*c^5*d*e - 4*b^4*c^3*d*e + 12*a^3*c^4*e*f + 6*b^5*c^2*d*f + 20*a*b^2*c^4*d*e
- 34*a*b^3*c^3*d*f + 29*a^2*b*c^4*d*f + 68*a*b^4*c^2*e*f - 76*a^2*b^2*c^3*e*f)/(4*a*c^7 - b^2*c^6) + (((b^3*c
^8)/2 - 2*a*b*c^9)*(3*b^5*f - 12*a^2*c^3*e + b^3*c^2*d - 2*b^4*c*e - 6*a*b*c^3*d - 20*a*b^3*c*f + 12*a*b^2*c^2
*e + 30*a^2*b*c^2*f)^2)/(c^8*(4*a*c - b^2)^3*(4*a*c^7 - b^2*c^6))))/(2*a*(4*a*c - b^2)^(3/2))) + (b*((((16*a^2
*c^5*f - 8*a*c^6*d + 16*a*b*c^5*e - 24*a*b^2*c^4*f)/c^6 - (8*a*c^2*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 25
6*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*
b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c^4*e))/(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^
2*b^2*c^6))*(6*b^8*f - 128*a^3*c^5*d + 2*b^6*c^2*d + 256*a^4*c^4*f - 4*b^7*c*e + 96*a^2*b^2*c^4*d - 192*a^2*b^
3*c^3*e + 336*a^2*b^4*c^2*f - 576*a^3*b^2*c^3*f - 76*a*b^6*c*f - 24*a*b^4*c^3*d + 48*a*b^5*c^2*e + 256*a^3*b*c
^4*e))/(2*(256*a^3*c^7 - 4*b^6*c^4 + 48*a*b^4*c^5 - 192*a^2*b^2*c^6)) - (a*c^4*d^2 + 9*a*b^4*f^2 + 4*a^3*c^2*f
^2 + 4*a*b^2*c^2*e^2 - 12*a^2*b^2*c*f^2 - 4*a^2*c^3*d*f + 6*a*b^2*c^2*d*f + 8*a^2*b*c^2*e*f - 4*a*b*c^3*d*e -
12*a*b^3*c*e*f)/c^6 + (a*(3*b^5*f - 12*a^2*c^3*e + b^3*c^2*d - 2*b^4*c*e - 6*a*b*c^3*d - 20*a*b^3*c*f + 12*a*b
^2*c^2*e + 30*a^2*b*c^2*f)^2)/(c^6*(4*a*c - b^2...
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