\(\int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx\) [9]

Optimal result

Integrand size = 32, antiderivative size = 545 \[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=-\frac {2 c d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {2 c d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {c e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {2+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {2 c f x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c f x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}-\frac {c g x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{n},\frac {4+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {c g x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{n},\frac {4+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )} \] Output:

-2*c*d*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(b^2- 
4*a*c-b*(-4*a*c+b^2)^(1/2))-2*c*d*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b 
+(-4*a*c+b^2)^(1/2)))/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)-c*e*x^2*hypergeom([ 
1, 2/n],[(2+n)/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(b^2-4*a*c-b*(-4*a*c+b^ 
2)^(1/2))-c*e*x^2*hypergeom([1, 2/n],[(2+n)/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1 
/2)))/(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)-2*c*f*x^3*hypergeom([1, 3/n],[(3+n) 
/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(3*b^2-12*a*c-3*b*(-4*a*c+b^2)^(1/2)) 
-2*c*f*x^3*hypergeom([1, 3/n],[(3+n)/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/( 
3*b^2-12*a*c+3*b*(-4*a*c+b^2)^(1/2))-c*g*x^4*hypergeom([1, 4/n],[(4+n)/n], 
-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(2*b^2-8*a*c-2*b*(-4*a*c+b^2)^(1/2))-c*g* 
x^4*hypergeom([1, 4/n],[(4+n)/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/(2*b^2-8 
*a*c+2*b*(-4*a*c+b^2)^(1/2))
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1093\) vs. \(2(545)=1090\).

Time = 2.32 (sec) , antiderivative size = 1093, normalized size of antiderivative = 2.01 \[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)),x]
 

Output:

(x*(3*g*x^3*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[- 
4/n, -4/n, (-4 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2* 
c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(4/n)) + (-b^2 + 4*a 
*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-4/n, -4/n, (-4 + n)/n, ( 
b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(2^(4/n)*((c*x^n 
)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(4/n)))) + 4*f*x^2*((-b^2 + 4*a*c - b 
*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b - Sq 
rt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[ 
b^2 - 4*a*c])/c + x^n))^(3/n)) + (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - 
 Hypergeometric2F1[-3/n, -3/n, (-3 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sq 
rt[b^2 - 4*a*c] + 2*c*x^n)]/(8^n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2* 
c*x^n))^(3/n)))) + 6*e*x*((-b^2 + 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hyperg 
eometric2F1[-2/n, -2/n, (-2 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 
- 4*a*c] + 2*c*x^n)]/(x^n/(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^(2/n)) 
+ (-b^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-2/n, -2/n, 
(-2 + n)/n, (b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(4^ 
n^(-1)*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^(2/n)))) + 12*d*((-b^2 
+ 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 - Hypergeometric2F1[-n^(-1), -n^(-1), (- 
1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)]/(x^n/ 
(-1/2*(-b + Sqrt[b^2 - 4*a*c])/c + x^n))^n^(-1)) - (Sqrt[b^2 - 4*a*c]*(...
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2325, 2432, 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.

\(\displaystyle \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx\)

\(\Big \downarrow \) 2325

\(\displaystyle \frac {2 c \int \frac {g x^3+f x^2+e x+d}{2 c x^n+b-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {g x^3+f x^2+e x+d}{2 c x^n+b+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 2432

\(\displaystyle \frac {2 c \int \left (-\frac {g x^3}{-2 c x^n-b+\sqrt {b^2-4 a c}}-\frac {f x^2}{-2 c x^n-b+\sqrt {b^2-4 a c}}-\frac {e x}{-2 c x^n-b+\sqrt {b^2-4 a c}}-\frac {d}{-2 c x^n-b+\sqrt {b^2-4 a c}}\right )dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \left (\frac {g x^3}{2 c x^n+b+\sqrt {b^2-4 a c}}+\frac {f x^2}{2 c x^n+b+\sqrt {b^2-4 a c}}+\frac {e x}{2 c x^n+b+\sqrt {b^2-4 a c}}+\frac {d}{2 c x^n+b+\sqrt {b^2-4 a c}}\right )dx}{\sqrt {b^2-4 a c}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 c \left (\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{b-\sqrt {b^2-4 a c}}+\frac {e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {n+2}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {f x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {g x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{n},\frac {n+4}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{4 \left (b-\sqrt {b^2-4 a c}\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}+b}+\frac {e x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2}{n},\frac {n+2}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 \left (\sqrt {b^2-4 a c}+b\right )}+\frac {f x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )}+\frac {g x^4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{n},\frac {n+4}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{4 \left (\sqrt {b^2-4 a c}+b\right )}\right )}{\sqrt {b^2-4 a c}}\)

Input:

Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)),x]
 

Output:

(2*c*((d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b 
^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c]) + (e*x^2*Hypergeometric2F1[1, 2/n, 
(2 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*(b - Sqrt[b^2 - 4*a*c]) 
) + (f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 
 4*a*c])])/(3*(b - Sqrt[b^2 - 4*a*c])) + (g*x^4*Hypergeometric2F1[1, 4/n, 
(4 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(4*(b - Sqrt[b^2 - 4*a*c]) 
)))/Sqrt[b^2 - 4*a*c] - (2*c*((d*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1) 
, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(b + Sqrt[b^2 - 4*a*c]) + (e*x^2*Hy 
pergeometric2F1[1, 2/n, (2 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2 
*(b + Sqrt[b^2 - 4*a*c])) + (f*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, (- 
2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(3*(b + Sqrt[b^2 - 4*a*c])) + (g*x^4*Hy 
pergeometric2F1[1, 4/n, (4 + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(4 
*(b + Sqrt[b^2 - 4*a*c]))))/Sqrt[b^2 - 4*a*c]
 

Defintions of rubi rules used

rule 2009

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

rule 2325

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int[Pq/(b - q + 2*c*x^n), x], x] - 
 Simp[2*(c/q)   Int[Pq/(b + q + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, n}, x] 
 && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0]
 

rule 2432

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ 
Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly 
Q[Pq, x^n])
 

Maple [F]

\[\int \frac {g \,x^{3}+f \,x^{2}+e x +d}{a +b \,x^{n}+c \,x^{2 n}}d x\]

Input:

int((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x)
 

Output:

int((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x)
 

Fricas [F]

\[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:

integrate((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")
 

Output:

integral((g*x^3 + f*x^2 + e*x + d)/(c*x^(2*n) + b*x^n + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\text {Timed out} \] Input:

integrate((g*x**3+f*x**2+e*x+d)/(a+b*x**n+c*x**(2*n)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:

integrate((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")
 

Output:

integrate((g*x^3 + f*x^2 + e*x + d)/(c*x^(2*n) + b*x^n + a), x)
 

Giac [F]

\[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{c x^{2 \, n} + b x^{n} + a} \,d x } \] Input:

integrate((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")
 

Output:

integrate((g*x^3 + f*x^2 + e*x + d)/(c*x^(2*n) + b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\int \frac {g\,x^3+f\,x^2+e\,x+d}{a+b\,x^n+c\,x^{2\,n}} \,d x \] Input:

int((d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)),x)
 

Output:

int((d + e*x + f*x^2 + g*x^3)/(a + b*x^n + c*x^(2*n)), x)
 

Reduce [F]

\[ \int \frac {d+e x+f x^2+g x^3}{a+b x^n+c x^{2 n}} \, dx=\left (\int \frac {x^{3}}{x^{2 n} c +x^{n} b +a}d x \right ) g +\left (\int \frac {x^{2}}{x^{2 n} c +x^{n} b +a}d x \right ) f +\left (\int \frac {x}{x^{2 n} c +x^{n} b +a}d x \right ) e +\left (\int \frac {1}{x^{2 n} c +x^{n} b +a}d x \right ) d \] Input:

int((g*x^3+f*x^2+e*x+d)/(a+b*x^n+c*x^(2*n)),x)
 

Output:

int(x**3/(x**(2*n)*c + x**n*b + a),x)*g + int(x**2/(x**(2*n)*c + x**n*b + 
a),x)*f + int(x/(x**(2*n)*c + x**n*b + a),x)*e + int(1/(x**(2*n)*c + x**n* 
b + a),x)*d