Integrand size = 20, antiderivative size = 173 \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d (1+m)}-\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m)} \] Output:
2*c*(d*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-2*c*x^2/(b-(-4*a*c+b ^2)^(1/2)))/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))/d/(1+m)-2*c*(d*x)^(1 +m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/ (-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))/d/(1+m)
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.47 \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\frac {(d x)^m \text {RootSum}\left [a+b \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{b \text {$\#$1}+2 c \text {$\#$1}^3}\&\right ]}{2 m} \] Input:
Integrate[(d*x)^m/(a + b*x^2 + c*x^4),x]
Output:
((d*x)^m*RootSum[a + b*#1^2 + c*#1^4 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1 + 2*c*#1^3)) & ])/(2*m)
Time = 0.55 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1451, 27, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx\) |
\(\Big \downarrow \) 1451 |
\(\displaystyle \frac {c \int \frac {2 (d x)^m}{2 c x^2+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {2 (d x)^m}{2 c x^2+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 c \int \frac {(d x)^m}{2 c x^2+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {(d x)^m}{2 c x^2+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}\) |
Input:
Int[(d*x)^m/(a + b*x^2 + c*x^4),x]
Output:
(2*c*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/( b - Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)) - (2*c*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, (-2*c *x^2)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]) *d*(1 + m))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[(d*x)^m/(b/2 - q/2 + c*x^2), x] , x] - Simp[c/q Int[(d*x)^m/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0]
\[\int \frac {\left (d x \right )^{m}}{c \,x^{4}+b \,x^{2}+a}d x\]
Input:
int((d*x)^m/(c*x^4+b*x^2+a),x)
Output:
int((d*x)^m/(c*x^4+b*x^2+a),x)
\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^4+b*x^2+a),x, algorithm="fricas")
Output:
integral((d*x)^m/(c*x^4 + b*x^2 + a), x)
\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int \frac {\left (d x\right )^{m}}{a + b x^{2} + c x^{4}}\, dx \] Input:
integrate((d*x)**m/(c*x**4+b*x**2+a),x)
Output:
Integral((d*x)**m/(a + b*x**2 + c*x**4), x)
\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^4+b*x^2+a),x, algorithm="maxima")
Output:
integrate((d*x)^m/(c*x^4 + b*x^2 + a), x)
\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^4+b*x^2+a),x, algorithm="giac")
Output:
integrate((d*x)^m/(c*x^4 + b*x^2 + a), x)
Timed out. \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int \frac {{\left (d\,x\right )}^m}{c\,x^4+b\,x^2+a} \,d x \] Input:
int((d*x)^m/(a + b*x^2 + c*x^4),x)
Output:
int((d*x)^m/(a + b*x^2 + c*x^4), x)
\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=d^{m} \left (\int \frac {x^{m}}{c \,x^{4}+b \,x^{2}+a}d x \right ) \] Input:
int((d*x)^m/(c*x^4+b*x^2+a),x)
Output:
d**m*int(x**m/(a + b*x**2 + c*x**4),x)