\(\int \frac {(d+e x^n)^3}{a+b x^n+c x^{2 n}} \, dx\) [63]

Optimal result

Integrand size = 26, antiderivative size = 345 \[ \int \frac {\left (d+e x^n\right )^3}{a+b x^n+c x^{2 n}} \, dx=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^{1+n}}{c (1+n)}-\frac {\left (b e^2 (3 c d-b e)-c e \left (3 c d^2-a e^2\right )-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{c^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {\left (2 c^3 d^3-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt {b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt {b^2-4 a c} e+3 b \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 2.77 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^n\right )^3}{a+b x^n+c x^{2 n}} \, dx=\frac {x \left (e^2 (3 c d-b e)+\frac {c e^3 x^n}{1+n}+\frac {\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b-\sqrt {b^2-4 a c}}+\frac {\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3+\frac {(-2 c d+b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b+\sqrt {b^2-4 a c}}\right )}{c^2} \] Input:

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

Output:

(x*(e^2*(3*c*d - b*e) + (c*e^3*x^n)/(1 + n) + ((3*c^2*d^2*e - 3*b*c*d*e^2 
+ b^2*e^3 - a*c*e^3 + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e 
)))/Sqrt[b^2 - 4*a*c])*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]) + ((3*c^2*d^2*e - 3*b*c 
*d*e^2 + b^2*e^3 - a*c*e^3 + ((-2*c*d + b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d 
 + 3*a*e)))/Sqrt[b^2 - 4*a*c])*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])))/c^2
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1754, 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.

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

\(\Big \downarrow \) 1754

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1754

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), 
 x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 
 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
 

rule 2009

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

Maple [F]

\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{a +b \,x^{n}+c \,x^{2 n}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(c*e^3*x*x^n + (3*c*d*e^2*(n + 1) - b*e^3*(n + 1))*x)/(c^2*(n + 1)) - inte 
grate(-(c^2*d^3 - (3*c*d*e^2 - b*e^3)*a + (3*c^2*d^2*e - 3*b*c*d*e^2 + b^2 
*e^3 - a*c*e^3)*x^n)/(c^3*x^(2*n) + b*c^2*x^n + a*c^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**n*b*e**3*x + int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*a*c*e**3*n + in 
t(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*a*c*e**3 - int(x**(2*n)/(x**(2*n)* 
c + x**n*b + a),x)*b**2*e**3*n - int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x) 
*b**2*e**3 + 3*int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*b*c*d*e**2*n + 3* 
int(x**(2*n)/(x**(2*n)*c + x**n*b + a),x)*b*c*d*e**2 - 3*int(x**(2*n)/(x** 
(2*n)*c + x**n*b + a),x)*c**2*d**2*e*n - 3*int(x**(2*n)/(x**(2*n)*c + x**n 
*b + a),x)*c**2*d**2*e + int(1/(x**(2*n)*c + x**n*b + a),x)*a**2*e**3*n + 
int(1/(x**(2*n)*c + x**n*b + a),x)*a**2*e**3 - 3*int(1/(x**(2*n)*c + x**n* 
b + a),x)*a*c*d**2*e*n - 3*int(1/(x**(2*n)*c + x**n*b + a),x)*a*c*d**2*e + 
 int(1/(x**(2*n)*c + x**n*b + a),x)*b*c*d**3*n + int(1/(x**(2*n)*c + x**n* 
b + a),x)*b*c*d**3 - a*e**3*n*x - a*e**3*x + 3*c*d**2*e*n*x + 3*c*d**2*e*x 
)/(b*c*(n + 1))