\(\int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx\) [72]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=-\frac {2 c (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) m}-\frac {2 c (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) m} \] Output:

-2*c*(d*x)^m*hypergeom([1, 1/2*m],[1+1/2*m],-2*c*x^2/(b-(-4*a*c+b^2)^(1/2) 
))/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/m-2*c*(d*x)^m*hypergeom([1, 1/2*m],[1+ 
1/2*m],-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/m
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.27 \[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\frac {(d x)^m \left (2 a \sqrt {b^2-4 a c} (2+m)-\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) m x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+\left (b^2-2 a c-b \sqrt {b^2-4 a c}\right ) m x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )\right )}{2 a^2 \sqrt {b^2-4 a c} m (2+m)} \] Input:

Integrate[(d*x)^m/(a*x + b*x^3 + c*x^5),x]
 

Output:

((d*x)^m*(2*a*Sqrt[b^2 - 4*a*c]*(2 + m) - (b^2 - 2*a*c + b*Sqrt[b^2 - 4*a* 
c])*m*x^2*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (2*c*x^2)/(-b + Sqrt[ 
b^2 - 4*a*c])] + (b^2 - 2*a*c - b*Sqrt[b^2 - 4*a*c])*m*x^2*Hypergeometric2 
F1[1, (2 + m)/2, (4 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]))/(2*a^2*S 
qrt[b^2 - 4*a*c]*m*(2 + m))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {9, 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.

\(\displaystyle \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle d \int \frac {(d x)^{m-1}}{c x^4+b x^2+a}dx\)

\(\Big \downarrow \) 1451

\(\displaystyle d \left (\frac {c \int \frac {2 (d x)^{m-1}}{2 c x^2+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {2 (d x)^{m-1}}{2 c x^2+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle d \left (\frac {2 c \int \frac {(d x)^{m-1}}{2 c x^2+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {(d x)^{m-1}}{2 c x^2+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\right )\)

\(\Big \downarrow \) 278

\(\displaystyle d \left (\frac {2 c (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{d m \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d m \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}\right )\)

Input:

Int[(d*x)^m/(a*x + b*x^3 + c*x^5),x]
 

Output:

d*((2*c*(d*x)^m*Hypergeometric2F1[1, m/2, (2 + 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*m) - (2*c*(d* 
x)^m*Hypergeometric2F1[1, m/2, (2 + 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*m))
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

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])
 

rule 1451

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]
 

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{c \,x^{5}+b \,x^{3}+x a}d x\]

Input:

int((d*x)^m/(c*x^5+b*x^3+a*x),x)
 

Output:

int((d*x)^m/(c*x^5+b*x^3+a*x),x)
 

Fricas [F]

\[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{5} + b x^{3} + a x} \,d x } \] Input:

integrate((d*x)^m/(c*x^5+b*x^3+a*x),x, algorithm="fricas")
 

Output:

integral((d*x)^m/(c*x^5 + b*x^3 + a*x), x)
 

Sympy [F]

\[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\int \frac {\left (d x\right )^{m}}{x \left (a + b x^{2} + c x^{4}\right )}\, dx \] Input:

integrate((d*x)**m/(c*x**5+b*x**3+a*x),x)
 

Output:

Integral((d*x)**m/(x*(a + b*x**2 + c*x**4)), x)
 

Maxima [F]

\[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{5} + b x^{3} + a x} \,d x } \] Input:

integrate((d*x)^m/(c*x^5+b*x^3+a*x),x, algorithm="maxima")
 

Output:

integrate((d*x)^m/(c*x^5 + b*x^3 + a*x), x)
 

Giac [F]

\[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{5} + b x^{3} + a x} \,d x } \] Input:

integrate((d*x)^m/(c*x^5+b*x^3+a*x),x, algorithm="giac")
 

Output:

integrate((d*x)^m/(c*x^5 + b*x^3 + a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=\int \frac {{\left (d\,x\right )}^m}{c\,x^5+b\,x^3+a\,x} \,d x \] Input:

int((d*x)^m/(a*x + b*x^3 + c*x^5),x)
 

Output:

int((d*x)^m/(a*x + b*x^3 + c*x^5), x)
 

Reduce [F]

\[ \int \frac {(d x)^m}{a x+b x^3+c x^5} \, dx=d^{m} \left (\int \frac {x^{m}}{c \,x^{5}+b \,x^{3}+a x}d x \right ) \] Input:

int((d*x)^m/(c*x^5+b*x^3+a*x),x)
 

Output:

d**m*int(x**m/(a*x + b*x**3 + c*x**5),x)