\(\int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx\) [85]

Optimal result

Integrand size = 28, antiderivative size = 175 \[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\frac {2 c x^{1-n} (d x)^m \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) (1+m-n)}-\frac {2 c x^{1-n} (d x)^m \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) (1+m-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.81 \[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\frac {x^{1-n} (d x)^m \left (\left (b+\sqrt {b^2-4 a c}\right ) \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\left (-b+\sqrt {b^2-4 a c}\right ) \operatorname {Hypergeometric2F1}\left (1,1+m-n,2+m-n,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )\right )}{2 a \sqrt {b^2-4 a c} (1+m-n)} \] Input:

Integrate[(d*x)^m/(a*x^n + b*x^(1 + n) + c*x^(2 + n)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2028, 30, 1150, 2009}

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.

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 1150

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d*x)^m/(a*x^n + b*x^(1 + n) + c*x^(2 + n)),x]
 

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 1150

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol 
] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + b*x + c*x^2), x], x] /; FreeQ[ 
{a, b, c, d, e, m}, x] &&  !IntegerQ[2*m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

Maple [F]

\[\int \frac {\left (d x \right )^{m}}{x^{n} a +b \,x^{1+n}+c \,x^{2+n}}d x\]

Input:

int((d*x)^m/(x^n*a+b*x^(1+n)+c*x^(2+n)),x)
 

Output:

int((d*x)^m/(x^n*a+b*x^(1+n)+c*x^(2+n)),x)
 

Fricas [F]

\[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{n + 2} + b x^{n + 1} + a x^{n}} \,d x } \] Input:

integrate((d*x)^m/(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="fricas")
 

Output:

integral((d*x)^m/(c*x^(n + 2) + b*x^(n + 1) + a*x^n), x)
 

Sympy [F]

\[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\int \frac {\left (d x\right )^{m}}{a x^{n} + b x^{n + 1} + c x^{n + 2}}\, dx \] Input:

integrate((d*x)**m/(a*x**n+b*x**(1+n)+c*x**(2+n)),x)
 

Output:

Integral((d*x)**m/(a*x**n + b*x**(n + 1) + c*x**(n + 2)), x)
 

Maxima [F]

\[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{n + 2} + b x^{n + 1} + a x^{n}} \,d x } \] Input:

integrate((d*x)^m/(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="maxima")
 

Output:

integrate((d*x)^m/(c*x^(n + 2) + b*x^(n + 1) + a*x^n), x)
 

Giac [F]

\[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{n + 2} + b x^{n + 1} + a x^{n}} \,d x } \] Input:

integrate((d*x)^m/(a*x^n+b*x^(1+n)+c*x^(2+n)),x, algorithm="giac")
 

Output:

integrate((d*x)^m/(c*x^(n + 2) + b*x^(n + 1) + a*x^n), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=\int \frac {{\left (d\,x\right )}^m}{a\,x^n+b\,x^{n+1}+c\,x^{n+2}} \,d x \] Input:

int((d*x)^m/(a*x^n + b*x^(n + 1) + c*x^(n + 2)),x)
 

Output:

int((d*x)^m/(a*x^n + b*x^(n + 1) + c*x^(n + 2)), x)
 

Reduce [F]

\[ \int \frac {(d x)^m}{a x^n+b x^{1+n}+c x^{2+n}} \, dx=d^{m} \left (\int \frac {x^{m}}{x^{n} a +x^{n} b x +x^{n} c \,x^{2}}d x \right ) \] Input:

int((d*x)^m/(a*x^n+b*x^(1+n)+c*x^(2+n)),x)
 

Output:

d**m*int(x**m/(x**n*a + x**n*b*x + x**n*c*x**2),x)