\(\int (a+b x^n)^2 (c+d x^n)^{-3-\frac {1}{n}} \, dx\) [128]

Optimal result

Integrand size = 25, antiderivative size = 127 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c (1+2 n)}-\frac {2 a (b c-a d) n x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {2 a n (b c+a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-2-\frac {1}{n}} \left (b^2 c^2 (1+n) x^{2 n}+2 a b c x^n \left (c+2 c n+d n x^n\right )+a^2 \left (c^2 \left (1+3 n+2 n^2\right )+2 c d n (1+2 n) x^n+2 d^2 n^2 x^{2 n}\right )\right )}{c^3 (1+n) (1+2 n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {903, 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.

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 746

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

Input:

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

Output:

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

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. \(587\) vs. \(2(127)=254\).

Time = 1.35 (sec) , antiderivative size = 588, normalized size of antiderivative = 4.63

method	result	size
parallelrisch	\(\frac {2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} d^{3} n^{2}+2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b c \,d^{2} n +x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{2} d n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c \,d^{2} n^{2}+x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{2} d +2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c \,d^{2} n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{2} d n +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{3} n +6 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d \,n^{2}+2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{2} d +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{3}+5 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d n +4 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{3} n +2 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3} n^{2}+x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{3}+3 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3} n +x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3}}{\left (1+n \right ) \left (1+2 n \right ) c^{3}}\)	\(588\)

Input:

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

Output:

(2*x*(x^n)^3*(c+d*x^n)^(-(1+3*n)/n)*a^2*d^3*n^2+2*x*(x^n)^3*(c+d*x^n)^(-(1 
+3*n)/n)*a*b*c*d^2*n+x*(x^n)^3*(c+d*x^n)^(-(1+3*n)/n)*b^2*c^2*d*n+6*x*(x^n 
)^2*(c+d*x^n)^(-(1+3*n)/n)*a^2*c*d^2*n^2+x*(x^n)^3*(c+d*x^n)^(-(1+3*n)/n)* 
b^2*c^2*d+2*x*(x^n)^2*(c+d*x^n)^(-(1+3*n)/n)*a^2*c*d^2*n+6*x*(x^n)^2*(c+d* 
x^n)^(-(1+3*n)/n)*a*b*c^2*d*n+x*(x^n)^2*(c+d*x^n)^(-(1+3*n)/n)*b^2*c^3*n+6 
*x*x^n*(c+d*x^n)^(-(1+3*n)/n)*a^2*c^2*d*n^2+2*x*(x^n)^2*(c+d*x^n)^(-(1+3*n 
)/n)*a*b*c^2*d+x*(x^n)^2*(c+d*x^n)^(-(1+3*n)/n)*b^2*c^3+5*x*x^n*(c+d*x^n)^ 
(-(1+3*n)/n)*a^2*c^2*d*n+4*x*x^n*(c+d*x^n)^(-(1+3*n)/n)*a*b*c^3*n+2*x*(c+d 
*x^n)^(-(1+3*n)/n)*a^2*c^3*n^2+x*x^n*(c+d*x^n)^(-(1+3*n)/n)*a^2*c^2*d+2*x* 
x^n*(c+d*x^n)^(-(1+3*n)/n)*a*b*c^3+3*x*(c+d*x^n)^(-(1+3*n)/n)*a^2*c^3*n+x* 
(c+d*x^n)^(-(1+3*n)/n)*a^2*c^3)/(1+n)/(1+2*n)/c^3
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.82 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {{\left (2 \, a^{2} d^{3} n^{2} + b^{2} c^{2} d + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} n\right )} x x^{3 \, n} + {\left (6 \, a^{2} c d^{2} n^{2} + b^{2} c^{3} + 2 \, a b c^{2} d + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a^{2} c^{2} d n^{2} + 2 \, a b c^{3} + a^{2} c^{2} d + {\left (4 \, a b c^{3} + 5 \, a^{2} c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a^{2} c^{3} n^{2} + 3 \, a^{2} c^{3} n + a^{2} c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )} {\left (d x^{n} + c\right )}^{\frac {3 \, n + 1}{n}}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (109) = 218\).

Time = 15.86 (sec) , antiderivative size = 1035, normalized size of antiderivative = 8.15 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Too large to display} \] Input:

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

Output:

2*a**2*c**2*c**(1/n)*c**(-3 - 2/n)*n**2*x*gamma(1/n)/(c**2*n**3*(1 + d*x** 
n/c)**(1/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 
 + 1/n) + d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 3*a** 
2*c**2*c**(1/n)*c**(-3 - 2/n)*n*x*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1 
/n)*gamma(3 + 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) 
+ d**2*n**3*x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + a**2*c**2*c** 
(1/n)*c**(-3 - 2/n)*x*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 
+ 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3* 
x**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 4*a**2*c*c**(1/n)*c**(-3 
- 2/n)*d*n**2*x*x**n*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 
 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3*x 
**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 2*a**2*c*c**(1/n)*c**(-3 - 
 2/n)*d*n*x*x**n*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n 
) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3*x**(2 
*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 2*a**2*c**(1/n)*c**(-3 - 2/n)* 
d**2*n**2*x*x**(2*n)*gamma(1/n)/(c**2*n**3*(1 + d*x**n/c)**(1/n)*gamma(3 + 
 1/n) + 2*c*d*n**3*x**n*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n) + d**2*n**3*x 
**(2*n)*(1 + d*x**n/c)**(1/n)*gamma(3 + 1/n)) + 4*a*b*c*c**(-3 - 1/n)*c**( 
1 + 1/n)*n*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c*d**(1 + 1/n)*n** 
2*gamma(3 + 1/n) + d*d**(1 + 1/n)*n**2*x**n*gamma(3 + 1/n)) + 2*a*b*c*c...
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((a+b*x^n)^2*(c+d*x^n)^(-3-1/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%%%{8,[1,0,4,3,1,3,2,0]%%%}+%%%{12,[1,0,4,2,1,3,2,0]%%%}+%%%{6 
,[1,0,4,1
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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