\(\int \frac {(c+d x^n)^{2-\frac {1}{n}}}{(a+b x^n)^3} \, dx\) [133]

Optimal result

Integrand size = 25, antiderivative size = 56 \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^3} \, dx=\frac {c^2 x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int \frac {\left (c+d x^n\right )^{2-\frac {1}{n}}}{\left (a+b x^n\right )^3} \, dx=\frac {c^2 x \left (c+d x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},\frac {(-b c+a d) x^n}{a \left (c+d x^n\right )}\right )}{a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {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.

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

\(\Big \downarrow \) 904

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

Input:

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

Output:

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

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

integrate((c+d*x**n)**(2-1/n)/(a+b*x**n)**3,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((c+d*x^n)^(2-1/n)/(a+b*x^n)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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