\(\int \frac {(c+d x^n)^{2-\frac {1}{n}}}{(a+b x^n)^4} \, dx\) [142]

Optimal result

Integrand size = 25, antiderivative size = 132 \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a (b c-a d) n \left (a+b x^n\right )^3}-\frac {c^2 \left (3 a d-b c \left (3-\frac {1}{n}\right )\right ) x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{3 a^4 (b c-a d)} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 37.05 (sec) , antiderivative size = 6405, normalized size of antiderivative = 48.52 \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {907, 904}

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.

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

\(\Big \downarrow \) 907

\(\displaystyle \frac {b x \left (c+d x^n\right )^{3-\frac {1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac {(3 a d n+b c (1-3 n)) \int \frac {\left (d x^n+c\right )^{2-\frac {1}{n}}}{\left (b x^n+a\right )^3}dx}{3 a n (b c-a d)}\)

\(\Big \downarrow \) 904

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 904

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*(x/(c^(p + 1)*(c + d*x^n)^(1/n)))*Hypergeometric2F1[1/n, -p, 1 
+ 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, q 
}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]
 

rule 907

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) 
  Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} 
, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  ! 
LtQ[q, -1]) && NeQ[p, -1]
 

Maple [F]

\[\int \frac {\left (c +d \,x^{n}\right )^{2-\frac {1}{n}}}{\left (a +b \,x^{n}\right )^{4}}d x\]

Input:

int((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x)
 

Output:

int((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x)
 

Fricas [F]

\[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 2}}{{\left (b x^{n} + a\right )}^{4}} \,d x } \] Input:

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x, algorithm="fricas")
 

Output:

integral((d*x^n + c)^((2*n - 1)/n)/(b^4*x^(4*n) + 4*a*b^3*x^(3*n) + 6*a^2* 
b^2*x^(2*n) + 4*a^3*b*x^n + a^4), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((c+d*x**n)**(2-1/n)/(a+b*x**n)**4,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 2}}{{\left (b x^{n} + a\right )}^{4}} \,d x } \] Input:

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^4, x)
 

Giac [F]

\[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 2}}{{\left (b x^{n} + a\right )}^{4}} \,d x } \] Input:

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x, algorithm="giac")
 

Output:

integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\int \frac {{\left (c+d\,x^n\right )}^{2-\frac {1}{n}}}{{\left (a+b\,x^n\right )}^4} \,d x \] Input:

int((c + d*x^n)^(2 - 1/n)/(a + b*x^n)^4,x)
 

Output:

int((c + d*x^n)^(2 - 1/n)/(a + b*x^n)^4, x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^4} \, dx=\left (\int \frac {x^{2 n}}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} b^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a \,b^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{2} b^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{3} b +\left (x^{n} d +c \right )^{\frac {1}{n}} a^{4}}d x \right ) d^{2}+2 \left (\int \frac {x^{n}}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} b^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a \,b^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{2} b^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{3} b +\left (x^{n} d +c \right )^{\frac {1}{n}} a^{4}}d x \right ) c d +\left (\int \frac {1}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} b^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a \,b^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{2} b^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} a^{3} b +\left (x^{n} d +c \right )^{\frac {1}{n}} a^{4}}d x \right ) c^{2} \] Input:

int((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x)
 

Output:

int(x**(2*n)/(x**(4*n)*(x**n*d + c)**(1/n)*b**4 + 4*x**(3*n)*(x**n*d + c)* 
*(1/n)*a*b**3 + 6*x**(2*n)*(x**n*d + c)**(1/n)*a**2*b**2 + 4*x**n*(x**n*d 
+ c)**(1/n)*a**3*b + (x**n*d + c)**(1/n)*a**4),x)*d**2 + 2*int(x**n/(x**(4 
*n)*(x**n*d + c)**(1/n)*b**4 + 4*x**(3*n)*(x**n*d + c)**(1/n)*a*b**3 + 6*x 
**(2*n)*(x**n*d + c)**(1/n)*a**2*b**2 + 4*x**n*(x**n*d + c)**(1/n)*a**3*b 
+ (x**n*d + c)**(1/n)*a**4),x)*c*d + int(1/(x**(4*n)*(x**n*d + c)**(1/n)*b 
**4 + 4*x**(3*n)*(x**n*d + c)**(1/n)*a*b**3 + 6*x**(2*n)*(x**n*d + c)**(1/ 
n)*a**2*b**2 + 4*x**n*(x**n*d + c)**(1/n)*a**3*b + (x**n*d + c)**(1/n)*a** 
4),x)*c**2