3.1.62 \(\int \frac {x^5 (d+e x^2+f x^4)}{(a+b x^2+c x^4)^2} \, dx\) [62]

Optimal. Leaf size=236 \[ \frac {\left (2 c^2 d+2 b^2 f-c (b e+6 a f)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (12 a^2 c^2 f-b^3 (c e-2 b f)-2 a c \left (2 c^2 d-3 b c e+6 b^2 f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {(c e-2 b f) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]

[Out]

1/2*(2*c^2*d+2*b^2*f-c*(6*a*f+b*e))*x^2/c^2/(-4*a*c+b^2)+1/2*x^4*(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*(12*a^2*c^2*f-b^3*(-2*b*f+c*e)-2*a*c*(6*b^2*f-3*b*c*e+2*c^2*d))
*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(3/2)+1/4*(-2*b*f+c*e)*ln(c*x^4+b*x^2+a)/c^3

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Rubi [A]
time = 0.29, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1677, 1658, 787, 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^2 f-2 a c \left (6 b^2 f-3 b c e+2 c^2 d\right )-\left (b^3 (c e-2 b f)\right )\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (-c (6 a f+b e)+2 b^2 f+2 c^2 d\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \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 {(c e-2 b f) \log \left (a+b x^2+c x^4\right )}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c^2*d + 2*b^2*f - c*(b*e + 6*a*f))*x^2)/(2*c^2*(b^2 - 4*a*c)) + (x^4*(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)) - ((12*a^2*c^2*f - b^3*(c*e - 2*b*f) -
2*a*c*(2*c^2*d - 3*b*c*e + 6*b^2*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) +
((c*e - 2*b*f)*Log[a + b*x^2 + c*x^4])/(4*c^3)

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 787

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 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^5 \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^2 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^4 \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 \left (2 \left (2 a e-\frac {b (c d+a f)}{c}\right )-\frac {\left (2 c^2 d-b c e+2 b^2 f-6 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 {\left (2 c^2 d+2 b^2 f-c (b e+6 a f)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \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 {\frac {a \left (2 c^2 d-b c e+2 b^2 f-6 a c f\right )}{c}+\left (\frac {b \left (2 c^2 d-b c e+2 b^2 f-6 a c f\right )}{c}+2 c \left (2 a e-\frac {b (c d+a f)}{c}\right )\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (b^2-4 a c\right )}\\ &=\frac {\left (2 c^2 d+2 b^2 f-c (b e+6 a f)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \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 {(c e-2 b f) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac {\left (12 a^2 c^2 f-b^3 (c e-2 b f)-2 a c \left (2 c^2 d-3 b c e+6 b^2 f\right )\right ) \text {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 )}\\ &=\frac {\left (2 c^2 d+2 b^2 f-c (b e+6 a f)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \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 {(c e-2 b f) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac {\left (12 a^2 c^2 f-b^3 (c e-2 b f)-2 a c \left (2 c^2 d-3 b c e+6 b^2 f\right )\right ) \text {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 )}\\ &=\frac {\left (2 c^2 d+2 b^2 f-c (b e+6 a f)\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \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 (12 a^2 c^2 f-b^3 (c e-2 b f)-2 a c \left (2 c^2 d-3 b c e+6 b^2 f\right )\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 )^{3/2}}+\frac {(c e-2 b f) \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 236, normalized size = 1.00 \begin {gather*} \frac {2 c f x^2-\frac {2 \left (b^2 \left (c^2 d-b c e+b^2 f\right ) x^2+a^2 c \left (-3 b f+2 c \left (e+f x^2\right )\right )+a \left (b^3 f-2 c^3 d x^2+b c^2 \left (d+3 e x^2\right )-b^2 c \left (e+4 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 (12 a^2 c^2 f+b^3 (-c e+2 b f)-2 a c \left (2 c^2 d-3 b c e+6 b^2 f\right )\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}}+(c e-2 b f) \log \left (a+b x^2+c x^4\right )}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*f*x^2 - (2*(b^2*(c^2*d - b*c*e + b^2*f)*x^2 + a^2*c*(-3*b*f + 2*c*(e + f*x^2)) + a*(b^3*f - 2*c^3*d*x^2 +
b*c^2*(d + 3*e*x^2) - b^2*c*(e + 4*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (2*(12*a^2*c^2*f + b^3*(-(
c*e) + 2*b*f) - 2*a*c*(2*c^2*d - 3*b*c*e + 6*b^2*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^
(3/2) + (c*e - 2*b*f)*Log[a + b*x^2 + c*x^4])/(4*c^3)

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Maple [A]
time = 0.10, size = 309, normalized size = 1.31

method	result	size
default	\(\frac {f \,x^{2}}{2 c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2} f -4 a \,b^{2} c f +3 a b \,c^{2} e -2 c^{3} a d +b^{4} f -b^{3} c e +b^{2} c^{2} d \right ) x^{2}}{c \left (4 a c -b^{2}\right )}+\frac {a \left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (8 a b c f -4 a \,c^{2} e -2 b^{3} f +b^{2} c e \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (6 a^{2} c f -2 a \,b^{2} f +a b c e -2 a \,c^{2} d -\frac {\left (8 a b c f -4 a \,c^{2} e -2 b^{3} f +b^{2} c e \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 c^{2}}\)	\(309\)
risch	\(\text {Expression too large to display}\)	\(3609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f*x^2/c^2-1/2/c^2*((-(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)
*x^2+a*(3*a*b*c*f-2*a*c^2*e-b^3*f+b^2*c*e-b*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*b*c*
f-4*a*c^2*e-2*b^3*f+b^2*c*e)/c*ln(c*x^4+b*x^2+a)+2*(6*a^2*c*f-2*a*b^2*f+a*b*c*e-2*a*c^2*d-1/2*(8*a*b*c*f-4*a*c
^2*e-2*b^3*f+b^2*c*e)*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^5*(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. 716 vs. \(2 (224) = 448\).
time = 0.46, size = 1455, normalized size = 6.17 \begin {gather*} \left [\frac {2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} f x^{6} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} f x^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} e + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} f\right )} x^{2} + {\left (4 \, a^{2} c^{3} d + {\left (4 \, a c^{4} d + {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} e - 2 \, {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} f\right )} x^{4} + {\left (4 \, a b c^{3} d + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} e - 2 \, {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} f\right )} x^{2} + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} e - 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2}\right )} f\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 ) - 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} e - 2 \, {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2}\right )} f + {\left ({\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e - 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} f\right )} x^{4} + {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e - 2 \, {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} f\right )} x^{2} + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} e - 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} f x^{6} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} f x^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} e + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} f\right )} x^{2} + 2 \, {\left (4 \, a^{2} c^{3} d + {\left (4 \, a c^{4} d + {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} e - 2 \, {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} f\right )} x^{4} + {\left (4 \, a b c^{3} d + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} e - 2 \, {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} f\right )} x^{2} + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} e - 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2}\right )} f\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 ) - 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} e - 2 \, {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2}\right )} f + {\left ({\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e - 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} f\right )} x^{4} + {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e - 2 \, {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} f\right )} x^{2} + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} e - 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*f*x^6 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*f*x^4 - 2*((b^4*c^
2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d - (b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c^3)*e + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 -
24*a^3*c^3)*f)*x^2 + (4*a^2*c^3*d + (4*a*c^4*d + (b^3*c^2 - 6*a*b*c^3)*e - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)
*f)*x^4 + (4*a*b*c^3*d + (b^4*c - 6*a*b^2*c^2)*e - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*f)*x^2 + (a*b^3*c - 6*a^2
*b*c^2)*e - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*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^3*c^2 - 4*a^2*b*c^3)*d + 2*(a*b^4*c - 6*a^2*b
^2*c^2 + 8*a^3*c^3)*e - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*f + (((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e -
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*f)*x^4 + ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e - 2*(b^6 - 8*a*b^4*c +
16*a^2*b^2*c^2)*f)*x^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*f)
*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 +
(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^2), 1/4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*f*x^6 + 2*(b^5*c - 8*
a*b^3*c^2 + 16*a^2*b*c^3)*f*x^4 - 2*((b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d - (b^5*c - 7*a*b^3*c^2 + 12*a^2*b*c
^3)*e + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*f)*x^2 + 2*(4*a^2*c^3*d + (4*a*c^4*d + (b^3*c^2 - 6*a*
b*c^3)*e - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*f)*x^4 + (4*a*b*c^3*d + (b^4*c - 6*a*b^2*c^2)*e - 2*(b^5 - 6*a*
b^3*c + 6*a^2*b*c^2)*f)*x^2 + (a*b^3*c - 6*a^2*b*c^2)*e - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*f)*sqrt(-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^3*c^2 - 4*a^2*b*c^3)*d + 2*(a*b^4*c - 6
*a^2*b^2*c^2 + 8*a^3*c^3)*e - 2*(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2)*f + (((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4
)*e - 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*f)*x^4 + ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e - 2*(b^6 - 8*a*b
^4*c + 16*a^2*b^2*c^2)*f)*x^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e - 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c
^2)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*
x^4 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*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**5*(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 = 5.88, size = 279, normalized size = 1.18 \begin {gather*} \frac {f x^{2}}{2 \, c^{2}} - \frac {{\left (4 \, a c^{3} d - 2 \, b^{4} f + 12 \, a b^{2} c f - 12 \, a^{2} c^{2} f + b^{3} c e - 6 \, a b c^{2} e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, b^{3} f x^{4} - 8 \, a b c f x^{4} - b^{2} c x^{4} e + 4 \, a c^{2} x^{4} e - 2 \, b^{2} c d x^{2} + 4 \, a c^{2} d x^{2} - 4 \, a^{2} c f x^{2} + b^{3} x^{2} e - 2 \, a b c x^{2} e - 2 \, a b c d - 2 \, a^{2} b f + a b^{2} e}{4 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac {{\left (2 \, b f - c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f*x^2/c^2 - 1/2*(4*a*c^3*d - 2*b^4*f + 12*a*b^2*c*f - 12*a^2*c^2*f + b^3*c*e - 6*a*b*c^2*e)*arctan((2*c*x^
2 + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + 1/4*(2*b^3*f*x^4 - 8*a*b*c*f*x^4 - b^2*c
*x^4*e + 4*a*c^2*x^4*e - 2*b^2*c*d*x^2 + 4*a*c^2*d*x^2 - 4*a^2*c*f*x^2 + b^3*x^2*e - 2*a*b*c*x^2*e - 2*a*b*c*d
- 2*a^2*b*f + a*b^2*e)/((c*x^4 + b*x^2 + a)*(b^2*c^2 - 4*a*c^3)) - 1/4*(2*b*f - c*e)*log(c*x^4 + b*x^2 + a)/c
^3

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Mupad [B]
time = 1.81, size = 2450, normalized size = 10.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*(b^3*f + 2*a*c^2*e + b*c^2*d - b^2*c*e - 3*a*b*c*f))/(2*c*(4*a*c - b^2)) + (x^2*(b^4*f + b^2*c^2*d + 2*a^2
*c^2*f - 2*a*c^3*d - b^3*c*e + 3*a*b*c^2*e - 4*a*b^2*c*f))/(2*c*(4*a*c - b^2)))/(a*c^2 + c^3*x^4 + b*c^2*x^2)
+ (f*x^2)/(2*c^2) + (log(a + b*x^2 + c*x^4)*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*
b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192
*a^2*b^2*c^5)) - (atan(((8*a*c^5*(4*a*c - b^2)^3 - 2*b^2*c^4*(4*a*c - b^2)^3)*(x^2*(((((24*a^2*c^5*f - 6*b^3*c
^4*e + 12*b^4*c^3*f - 8*a*c^6*d + 28*a*b*c^5*e - 56*a*b^2*c^4*f)/(4*a*c^5 - b^2*c^4) + ((8*b^3*c^6 - 32*a*b*c^
7)*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e
- 256*a^3*b*c^3*f))/(2*(4*a*c^5 - b^2*c^4)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)))*(2*b^
4*f + 12*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a*b*c^2*e - 12*a*b^2*c*f))/(8*c^3*(4*a*c - b^2)^(3/2)) + ((8*b^3*
c^6 - 32*a*b*c^7)*(2*b^4*f + 12*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a*b*c^2*e - 12*a*b^2*c*f)*(4*b^7*f + 128*a
^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f)
)/(16*c^3*(4*a*c - b^2)^(3/2)*(4*a*c^5 - b^2*c^4)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)))
/(a*(4*a*c - b^2)) + (b*((4*b^5*f^2 + b^3*c^2*e^2 + 12*a^2*b*c^2*f^2 + 2*a*c^4*d*e - 4*b^4*c*e*f - 5*a*b*c^3*e
^2 - 20*a*b^3*c*f^2 - 6*a^2*c^3*e*f + 20*a*b^2*c^2*e*f - 4*a*b*c^3*d*f)/(4*a*c^5 - b^2*c^4) + (((24*a^2*c^5*f
- 6*b^3*c^4*e + 12*b^4*c^3*f - 8*a*c^6*d + 28*a*b*c^5*e - 56*a*b^2*c^4*f)/(4*a*c^5 - b^2*c^4) + ((8*b^3*c^6 -
32*a*b*c^7)*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*
b^4*c^2*e - 256*a^3*b*c^3*f))/(2*(4*a*c^5 - b^2*c^4)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5
)))*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*
e - 256*a^3*b*c^3*f))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5)) - (((b^3*c^6)/2 - 2*a*b*c
^7)*(2*b^4*f + 12*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a*b*c^2*e - 12*a*b^2*c*f)^2)/(c^6*(4*a*c - b^2)^3*(4*a*c
^5 - b^2*c^4))))/(2*a*(4*a*c - b^2)^(3/2))) + ((((8*a*c^4*e - 16*a*b*c^3*f)/c^4 - (8*a*c^2*(4*b^7*f + 128*a^3*
c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f))/(
256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5))*(2*b^4*f + 12*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a
*b*c^2*e - 12*a*b^2*c*f))/(8*c^3*(4*a*c - b^2)^(3/2)) - (a*(2*b^4*f + 12*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a
*b*c^2*e - 12*a*b^2*c*f)*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^
5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f))/(c*(4*a*c - b^2)^(3/2)*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 19
2*a^2*b^2*c^5)))/(a*(4*a*c - b^2)) + (b*((((8*a*c^4*e - 16*a*b*c^3*f)/c^4 - (8*a*c^2*(4*b^7*f + 128*a^3*c^4*e
- 2*b^6*c*e - 96*a^2*b^2*c^3*e + 192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f))/(256*a^
3*c^6 - 4*b^6*c^3 + 48*a*b^4*c^4 - 192*a^2*b^2*c^5))*(4*b^7*f + 128*a^3*c^4*e - 2*b^6*c*e - 96*a^2*b^2*c^3*e +
192*a^2*b^3*c^2*f - 48*a*b^5*c*f + 24*a*b^4*c^2*e - 256*a^3*b*c^3*f))/(2*(256*a^3*c^6 - 4*b^6*c^3 + 48*a*b^4*
c^4 - 192*a^2*b^2*c^5)) - (4*a*b^2*f^2 + a*c^2*e^2 - 4*a*b*c*e*f)/c^4 + (a*(2*b^4*f + 12*a^2*c^2*f - 4*a*c^3*d
- b^3*c*e + 6*a*b*c^2*e - 12*a*b^2*c*f)^2)/(c^4*(4*a*c - b^2)^3)))/(2*a*(4*a*c - b^2)^(3/2))))/(4*b^8*f^2 + 1
6*a^2*c^6*d^2 + 144*a^4*c^4*f^2 + b^6*c^2*e^2 - 12*a*b^4*c^3*e^2 - 4*b^7*c*e*f + 36*a^2*b^2*c^4*e^2 + 192*a^2*
b^4*c^2*f^2 - 288*a^3*b^2*c^3*f^2 - 48*a*b^6*c*f^2 - 96*a^3*c^5*d*f + 8*a*b^3*c^4*d*e - 48*a^2*b*c^5*d*e - 16*
a*b^4*c^3*d*f + 48*a*b^5*c^2*e*f + 144*a^3*b*c^4*e*f + 96*a^2*b^2*c^4*d*f - 168*a^2*b^3*c^3*e*f))*(2*b^4*f + 1
2*a^2*c^2*f - 4*a*c^3*d - b^3*c*e + 6*a*b*c^2*e - 12*a*b^2*c*f))/(2*c^3*(4*a*c - b^2)^(3/2))

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