Integrand size = 20, antiderivative size = 173 \[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{3},\frac {4+m}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d (1+m)}-\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{3},\frac {4+m}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m)} \] Output:
2*c*(d*x)^(1+m)*hypergeom([1, 1/3+1/3*m],[4/3+1/3*m],-2*c*x^3/(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, 1/3+1/3*m],[4/3+1/3*m],-2*c*x^3/(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 = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49 \[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\frac {(d x)^m \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ]}{3 m} \] Input:
Integrate[(d*x)^m/(a + b*x^3 + c*x^6),x]
Output:
((d*x)^m*RootSum[a + b*#1^3 + c*#1^6 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(b*#1^2 + 2*c*#1^5)) & ])/(3*m)
Time = 0.33 (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 = {1711, 27, 888}
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.
\(\displaystyle \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx\) |
\(\Big \downarrow \) 1711 |
\(\displaystyle \frac {c \int \frac {2 (d x)^m}{2 c x^3+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {2 (d x)^m}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 c \int \frac {(d x)^m}{2 c x^3+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {(d x)^m}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{3},\frac {m+4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{3},\frac {m+4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}\) |
Input:
Int[(d*x)^m/(a + b*x^3 + c*x^6),x]
Output:
(2*c*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, (-2*c*x^3)/( 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)/3, (4 + m)/3, (-2*c *x^3)/(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_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symb ol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[(d*x)^m/(b/2 - q/2 + c *x^n), x], x] - Simp[c/q Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; Free Q[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
\[\int \frac {\left (d x \right )^{m}}{c \,x^{6}+b \,x^{3}+a}d x\]
Input:
int((d*x)^m/(c*x^6+b*x^3+a),x)
Output:
int((d*x)^m/(c*x^6+b*x^3+a),x)
\[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
integral((d*x)^m/(c*x^6 + b*x^3 + a), x)
Timed out. \[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\text {Timed out} \] Input:
integrate((d*x)**m/(c*x**6+b*x**3+a),x)
Output:
Timed out
\[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
integrate((d*x)^m/(c*x^6 + b*x^3 + a), x)
\[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate((d*x)^m/(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
integrate((d*x)^m/(c*x^6 + b*x^3 + a), x)
Timed out. \[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=\int \frac {{\left (d\,x\right )}^m}{c\,x^6+b\,x^3+a} \,d x \] Input:
int((d*x)^m/(a + b*x^3 + c*x^6),x)
Output:
int((d*x)^m/(a + b*x^3 + c*x^6), x)
\[ \int \frac {(d x)^m}{a+b x^3+c x^6} \, dx=d^{m} \left (\int \frac {x^{m}}{c \,x^{6}+b \,x^{3}+a}d x \right ) \] Input:
int((d*x)^m/(c*x^6+b*x^3+a),x)
Output:
d**m*int(x**m/(a + b*x**3 + c*x**6),x)