Integrand size = 23, antiderivative size = 127 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=-\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {n (b c+2 a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)} \] Output:
-(-a*d+b*c)*x*(c+d*x^n)^(-2-1/n)/c/d/(1+2*n)+(2*a*d*n+b*c)*x*(c+d*x^n)^(-1 -1/n)/c^2/d/(1+n)/(1+2*n)+n*(2*a*d*n+b*c)*x/c^3/d/(1+n)/(1+2*n)/((c+d*x^n) ^(1/n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (b c \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )+(-b c+a d) \operatorname {Hypergeometric2F1}\left (3+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )}{c^3 d} \] Input:
Integrate[(a + b*x^n)*(c + d*x^n)^(-3 - n^(-1)),x]
Output:
(x*(1 + (d*x^n)/c)^n^(-1)*(b*c*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n ^(-1), -((d*x^n)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[3 + n^(-1), n^(-1) , 1 + n^(-1), -((d*x^n)/c)]))/(c^3*d*(c + d*x^n)^n^(-1))
Time = 0.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {910, 777, 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.
\(\displaystyle \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-3} \, dx\) |
\(\Big \downarrow \) 910 |
\(\displaystyle \frac {(2 a d n+b c) \int \left (d x^n+c\right )^{-2-\frac {1}{n}}dx}{c d (2 n+1)}-\frac {x (b c-a d) \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c d (2 n+1)}\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \frac {(2 a d n+b c) \left (\frac {n \int \left (d x^n+c\right )^{-1-\frac {1}{n}}dx}{c (n+1)}+\frac {x \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}\right )}{c d (2 n+1)}-\frac {x (b c-a d) \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c d (2 n+1)}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {\left (\frac {n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}+\frac {x \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}\right ) (2 a d n+b c)}{c d (2 n+1)}-\frac {x (b c-a d) \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c d (2 n+1)}\) |
Input:
Int[(a + b*x^n)*(c + d*x^n)^(-3 - n^(-1)),x]
Output:
-(((b*c - a*d)*x*(c + d*x^n)^(-2 - n^(-1)))/(c*d*(1 + 2*n))) + ((b*c + 2*a *d*n)*((x*(c + d*x^n)^(-1 - n^(-1)))/(c*(1 + n)) + (n*x)/(c^2*(1 + n)*(c + d*x^n)^n^(-1))))/(c*d*(1 + 2*n))
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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1], 0] && NeQ[p, -1]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(127)=254\).
Time = 1.00 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.42
method | result | size |
parallelrisch | \(\frac {2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,d^{3} n^{2}+x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b c \,d^{2} n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a c \,d^{2} n^{2}+2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a c \,d^{2} n +3 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{2} d n +6 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d \,n^{2}+x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{2} d +5 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d n +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{3} n +2 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3} n^{2}+x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{2} d +x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b \,c^{3}+3 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3} n +x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a \,c^{3}}{\left (1+n \right ) \left (1+2 n \right ) c^{3}}\) | \(434\) |
Input:
int((a+b*x^n)*(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*d^3*n^2+x*(x^n)^3*(c+d*x^n)^(-(1+3*n )/n)*b*c*d^2*n+6*x*(x^n)^2*(c+d*x^n)^(-(1+3*n)/n)*a*c*d^2*n^2+2*x*(x^n)^2* (c+d*x^n)^(-(1+3*n)/n)*a*c*d^2*n+3*x*(x^n)^2*(c+d*x^n)^(-(1+3*n)/n)*b*c^2* d*n+6*x*x^n*(c+d*x^n)^(-(1+3*n)/n)*a*c^2*d*n^2+x*(x^n)^2*(c+d*x^n)^(-(1+3* n)/n)*b*c^2*d+5*x*x^n*(c+d*x^n)^(-(1+3*n)/n)*a*c^2*d*n+2*x*x^n*(c+d*x^n)^( -(1+3*n)/n)*b*c^3*n+2*x*(c+d*x^n)^(-(1+3*n)/n)*a*c^3*n^2+x*x^n*(c+d*x^n)^( -(1+3*n)/n)*a*c^2*d+x*x^n*(c+d*x^n)^(-(1+3*n)/n)*b*c^3+3*x*(c+d*x^n)^(-(1+ 3*n)/n)*a*c^3*n+x*(c+d*x^n)^(-(1+3*n)/n)*a*c^3)/(1+n)/(1+2*n)/c^3
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.36 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {{\left (2 \, a d^{3} n^{2} + b c d^{2} n\right )} x x^{3 \, n} + {\left (6 \, a c d^{2} n^{2} + b c^{2} d + {\left (3 \, b c^{2} d + 2 \, a c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a c^{2} d n^{2} + b c^{3} + a c^{2} d + {\left (2 \, b c^{3} + 5 \, a c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a c^{3} n^{2} + 3 \, a c^{3} n + a c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )} {\left (d x^{n} + c\right )}^{\frac {3 \, n + 1}{n}}} \] Input:
integrate((a+b*x^n)*(c+d*x^n)^(-3-1/n),x, algorithm="fricas")
Output:
((2*a*d^3*n^2 + b*c*d^2*n)*x*x^(3*n) + (6*a*c*d^2*n^2 + b*c^2*d + (3*b*c^2 *d + 2*a*c*d^2)*n)*x*x^(2*n) + (6*a*c^2*d*n^2 + b*c^3 + a*c^2*d + (2*b*c^3 + 5*a*c^2*d)*n)*x*x^n + (2*a*c^3*n^2 + 3*a*c^3*n + a*c^3)*x)/((2*c^3*n^2 + 3*c^3*n + c^3)*(d*x^n + c)^((3*n + 1)/n))
Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (105) = 210\).
Time = 2.49 (sec) , antiderivative size = 959, normalized size of antiderivative = 7.55 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**n)*(c+d*x**n)**(-3-1/n),x)
Output:
2*a*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*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)*ga mma(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*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*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*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*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)) + 2*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)) + b*c*c**(-3 - 1/n)*c**(1 + 1/n...
\[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 3} \,d x } \] Input:
integrate((a+b*x^n)*(c+d*x^n)^(-3-1/n),x, algorithm="maxima")
Output:
integrate((b*x^n + a)*(d*x^n + c)^(-1/n - 3), x)
Exception generated. \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*x^n)*(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%%%{4,[0,0,2,2,1,1,0,1]%%%}+%%%{2,[0,0,2,1,1,1,0,1]%%%}+%%%{2, [0,0,2,1,
Timed out. \[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+3}} \,d x \] Input:
int((a + b*x^n)/(c + d*x^n)^(1/n + 3),x)
Output:
int((a + b*x^n)/(c + d*x^n)^(1/n + 3), x)
\[ \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\left (\int \frac {x^{n}}{x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{3}+3 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{2}+3 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{3}}d x \right ) b +\left (\int \frac {1}{x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{3}+3 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{2}+3 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{3}}d x \right ) a \] Input:
int((a+b*x^n)*(c+d*x^n)^(-3-1/n),x)
Output:
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)*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