\(\int (a+b x^n) (c+d x^n)^{-2-\frac {1}{n}} \, dx\) [129]

Optimal result

Integrand size = 23, antiderivative size = 72 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=-\frac {(b c-a d) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c d (1+n)}+\frac {(b c+a d n) x \left (c+d x^n\right )^{-1/n}}{c^2 d (1+n)} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-\frac {1+n}{n}} \left (b c x^n+a (1+n) \left (c+d x^n\right ) \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{c^2 (1+n)} \] Input:

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

Output:

(x*(b*c*x^n + a*(1 + n)*(c + d*x^n)*(1 + (d*x^n)/c)^n^(-1)*Hypergeometric2 
F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)]))/(c^2*(1 + n)*(c + d*x^n 
)^((1 + n)/n))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {903, 746}

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.

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 746

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 746

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) 
/a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
 

rule 903

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(72)=144\).

Time = 0.99 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.76

method	result	size
parallelrisch	\(\frac {x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,d^{2} n +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} b c d +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a c d n +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a c d +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} b \,c^{2}+x \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,c^{2} n +x \left (c +d \,x^{n}\right )^{-\frac {1+2 n}{n}} a \,c^{2}}{c^{2} \left (1+n \right )}\)	\(199\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.18 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {{\left (a d^{2} n + b c d\right )} x x^{2 \, n} + {\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} + {\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )} {\left (d x^{n} + c\right )}^{\frac {2 \, n + 1}{n}}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (56) = 112\).

Time = 2.27 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.32 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx=\frac {a c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} n \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a c c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {a c^{\frac {1}{n}} c^{-2 - \frac {1}{n}} d n x^{n} \Gamma \left (\frac {1}{n}\right )}{c d^{\frac {1}{n}} n^{2} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right ) + d d^{\frac {1}{n}} n^{2} x^{n} \left (\frac {c x^{- n}}{d} + 1\right )^{\frac {1}{n}} \Gamma \left (2 + \frac {1}{n}\right )} + \frac {b c^{-2 - \frac {1}{n}} c^{1 + \frac {1}{n}} d^{-1 - \frac {1}{n}} \left (\frac {c x^{- n}}{d} + 1\right )^{-1 - \frac {1}{n}} \Gamma \left (1 + \frac {1}{n}\right )}{n \Gamma \left (2 + \frac {1}{n}\right )} \] Input:

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

Output:

a*c*c**(1/n)*c**(-2 - 1/n)*n*gamma(1/n)/(c*d**(1/n)*n**2*(c/(d*x**n) + 1)* 
*(1/n)*gamma(2 + 1/n) + d*d**(1/n)*n**2*x**n*(c/(d*x**n) + 1)**(1/n)*gamma 
(2 + 1/n)) + a*c*c**(1/n)*c**(-2 - 1/n)*gamma(1/n)/(c*d**(1/n)*n**2*(c/(d* 
x**n) + 1)**(1/n)*gamma(2 + 1/n) + d*d**(1/n)*n**2*x**n*(c/(d*x**n) + 1)** 
(1/n)*gamma(2 + 1/n)) + a*c**(1/n)*c**(-2 - 1/n)*d*n*x**n*gamma(1/n)/(c*d* 
*(1/n)*n**2*(c/(d*x**n) + 1)**(1/n)*gamma(2 + 1/n) + d*d**(1/n)*n**2*x**n* 
(c/(d*x**n) + 1)**(1/n)*gamma(2 + 1/n)) + b*c**(-2 - 1/n)*c**(1 + 1/n)*d** 
(-1 - 1/n)*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(n*gamma(2 + 1/n))
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,0,2,2,1,1,0,1]%%%}+%%%{1,[0,0,2,1,1,1,0,1]%%%}+%%%{1, 
[0,0,2,1,
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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