3.41 $$\int \frac{(d x)^m (A+B x+C x^2)}{(a+b x^2+c x^4)^2} \, dx$$

Optimal. Leaf size=685 $\frac{c (d x)^{m+1} \left (A \left (b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left (2 b-(1-m) \sqrt{b^2-4 a c}\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (A \left (-b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left ((1-m) \sqrt{b^2-4 a c}+2 b\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B c (d x)^{m+2} \left (b m \left (\sqrt{b^2-4 a c}+b\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{B c (d x)^{m+2} \left (b m \left (b-\sqrt{b^2-4 a c}\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{B (d x)^{m+2} \left (-2 a c+b^2+b c x^2\right )}{2 a d^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}$

[Out]

(B*(d*x)^(2 + m)*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*d^2*(a + b*x^2 + c*x^4)) + ((d*x)^(1 + m)*(A*(b^2
- 2*a*c) - a*b*C + c*(A*b - 2*a*C)*x^2))/(2*a*(b^2 - 4*a*c)*d*(a + b*x^2 + c*x^4)) + (c*(2*a*C*(2*b - Sqrt[b^
2 - 4*a*c]*(1 - m)) + A*(b^2*(1 - m) + b*Sqrt[b^2 - 4*a*c]*(1 - m) - 4*a*c*(3 - m)))*(d*x)^(1 + m)*Hypergeomet
ric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 -
4*a*c])*d*(1 + m)) - (c*(2*a*C*(2*b + Sqrt[b^2 - 4*a*c]*(1 - m)) + A*(b^2*(1 - m) - b*Sqrt[b^2 - 4*a*c]*(1 - m
) - 4*a*c*(3 - m)))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]
)])/(2*a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m)) - (B*c*(4*a*c*(2 - m) + b*(b + Sqrt[b^2 - 4*a*
c])*m)*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/(2*a*(b^2
- 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d^2*(2 + m)) + (B*c*(4*a*c*(2 - m) + b*(b - Sqrt[b^2 - 4*a*c])*m)*(d*x
)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)^(
3/2)*(b + Sqrt[b^2 - 4*a*c])*d^2*(2 + m))

________________________________________________________________________________________

Rubi [A]  time = 2.37779, antiderivative size = 670, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 6, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {1662, 1277, 1285, 364, 12, 1121} $\frac{c (d x)^{m+1} \left (A \left (b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left (2 b-(1-m) \sqrt{b^2-4 a c}\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (-(1-m) \sqrt{b^2-4 a c} (A b-2 a C)-4 a A c (3-m)+4 a b C+A b^2 (1-m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B c (d x)^{m+2} \left (b m \left (\sqrt{b^2-4 a c}+b\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{B c (d x)^{m+2} \left (b m \left (b-\sqrt{b^2-4 a c}\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{B (d x)^{m+2} \left (-2 a c+b^2+b c x^2\right )}{2 a d^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[((d*x)^m*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(B*(d*x)^(2 + m)*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*d^2*(a + b*x^2 + c*x^4)) + ((d*x)^(1 + m)*(A*(b^2
- 2*a*c) - a*b*C + c*(A*b - 2*a*C)*x^2))/(2*a*(b^2 - 4*a*c)*d*(a + b*x^2 + c*x^4)) + (c*(2*a*C*(2*b - Sqrt[b^
2 - 4*a*c]*(1 - m)) + A*(b^2*(1 - m) + b*Sqrt[b^2 - 4*a*c]*(1 - m) - 4*a*c*(3 - m)))*(d*x)^(1 + m)*Hypergeomet
ric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 -
4*a*c])*d*(1 + m)) - (c*(4*a*b*C + A*b^2*(1 - m) - Sqrt[b^2 - 4*a*c]*(A*b - 2*a*C)*(1 - m) - 4*a*A*c*(3 - m))*
(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*
c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m)) - (B*c*(4*a*c*(2 - m) + b*(b + Sqrt[b^2 - 4*a*c])*m)*(d*x)^(2 + m)
*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)^(3/2)*(b -
Sqrt[b^2 - 4*a*c])*d^2*(2 + m)) + (B*c*(4*a*c*(2 - m) + b*(b - Sqrt[b^2 - 4*a*c])*m)*(d*x)^(2 + m)*Hypergeome
tric2F1[1, (2 + m)/2, (4 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(2*a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 -
4*a*c])*d^2*(2 + m))

Rule 1662

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x],
k}, Int[(d*x)^m*Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(a + b*x^2 + c*x^4)^p, x] + Dist[1/d, Int[(d*
x)^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2 + 1}]*(a + b*x^2 + c*x^4)^p, x], x]] /; FreeQ[{
a, b, c, d, m, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1277

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

Rule 1285

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt
[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d
- b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c,
0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1121

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[((d*x)^(m + 1)*(b^2 - 2*a
*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*d*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*
c)), Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7)*
x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (Integer
Q[p] || IntegerQ[m])

Rubi steps

\begin{align*} \int \frac{(d x)^m \left (A+B x+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{\int \frac{B (d x)^{1+m}}{\left (a+b x^2+c x^4\right )^2} \, dx}{d}+\int \frac{(d x)^m \left (A+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}-\frac{\int \frac{(d x)^m \left (-A b^2 (1-m)+2 a A c (3-m)-a b C (1+m)-c (A b-2 a C) (1-m) x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}+\frac{B \int \frac{(d x)^{1+m}}{\left (a+b x^2+c x^4\right )^2} \, dx}{d}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}-\frac{B \int \frac{(d x)^{1+m} \left (2 a c (2-m)+b^2 m+b c m x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right ) d}-\frac{\left (c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right )\right ) \int \frac{(d x)^m}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2}}+\frac{\left (c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right )\right ) \int \frac{(d x)^m}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}+\frac{c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{\left (B c \left (4 a c (2-m)+b \left (b-\sqrt{b^2-4 a c}\right ) m\right )\right ) \int \frac{(d x)^{1+m}}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2} d}-\frac{\left (B c \left (4 a c (2-m)+b \left (b+\sqrt{b^2-4 a c}\right ) m\right )\right ) \int \frac{(d x)^{1+m}}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )^{3/2} d}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}+\frac{c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{B c \left (4 a c (2-m)+b \left (b+\sqrt{b^2-4 a c}\right ) m\right ) (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d^2 (2+m)}+\frac{B c \left (4 a c (2-m)+b \left (b-\sqrt{b^2-4 a c}\right ) m\right ) (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d^2 (2+m)}\\ \end{align*}

Mathematica [C]  time = 0.344501, size = 242, normalized size = 0.35 $\frac{x (d x)^m \left (A \left (m^2+5 m+6\right ) F_1\left (\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+(m+1) x \left (B (m+3) F_1\left (\frac{m+2}{2};2,2;\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+C (m+2) x F_1\left (\frac{m+3}{2};2,2;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}{a^2 (m+1) (m+2) (m+3)}$

Warning: Unable to verify antiderivative.

[In]

Integrate[((d*x)^m*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(d*x)^m*(A*(6 + 5*m + m^2)*AppellF1[(1 + m)/2, 2, 2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^
2)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*x*(B*(3 + m)*AppellF1[(2 + m)/2, 2, 2, (4 + m)/2, (-2*c*x^2)/(b + Sqrt[
b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + C*(2 + m)*x*AppellF1[(3 + m)/2, 2, 2, (5 + m)/2, (-2*c*x^
2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])))/(a^2*(1 + m)*(2 + m)*(3 + m))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m} \left ( C{x}^{2}+Bx+A \right ) }{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x)

[Out]

int((d*x)^m*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*(d*x)^m/(c*x^4 + b*x^2 + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{c^{2} x^{8} + 2 \, b c x^{6} +{\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*(d*x)^m/(c^2*x^8 + 2*b*c*x^6 + (b^2 + 2*a*c)*x^4 + 2*a*b*x^2 + a^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(C*x**2+B*x+A)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*(d*x)^m/(c*x^4 + b*x^2 + a)^2, x)