Integrand size = 25, antiderivative size = 327 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{3 a (b c-a d) n}-\frac {(3 a d n-b (c+3 c n)) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{3 a c (b c-a d) n (1+3 n)}-\frac {(3 a d n-b (c+3 c n)) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 (b c-a d) \left (1+5 n+6 n^2\right )}-\frac {2 a n (3 a d n-b (c+3 c n)) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 (b c-a d) (1+n) (1+2 n) (1+3 n)}-\frac {2 a^2 n^2 (3 a d n-b (c+3 c n)) x \left (c+d x^n\right )^{-1/n}}{c^4 (b c-a d) (1+n) (1+2 n) (1+3 n)} \]
-1/3*b*x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/a/(-a*d+b*c)/n-1/3*(3*a*d*n-b*(3*c *n+c))*x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/a/c/(-a*d+b*c)/n/(1+3*n)-(3*a*d*n- b*(3*c*n+c))*x*(a+b*x^n)^2*(c+d*x^n)^(-2-1/n)/c^2/(-a*d+b*c)/(6*n^2+5*n+1) -2*a*n*(3*a*d*n-b*(3*c*n+c))*x*(a+b*x^n)*(c+d*x^n)^(-1-1/n)/c^3/(-a*d+b*c) /(6*n^3+11*n^2+6*n+1)-2*a^2*n^2*(3*a*d*n-b*(3*c*n+c))*x/c^4/(-a*d+b*c)/(6* n^3+11*n^2+6*n+1)/((c+d*x^n)^(1/n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.44 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.42 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (b^2 c^2 \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )-(b c-a d) \left (2 b c \operatorname {Hypergeometric2F1}\left (3+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )+(-b c+a d) \operatorname {Hypergeometric2F1}\left (4+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )\right )}{c^4 d^2} \]
(x*(1 + (d*x^n)/c)^n^(-1)*(b^2*c^2*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)] - (b*c - a*d)*(2*b*c*Hypergeometric2F1[3 + n^(-1) , n^(-1), 1 + n^(-1), -((d*x^n)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[4 + n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)])))/(c^4*d^2*(c + d*x^n)^n^(-1))
Time = 0.38 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {907, 903, 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.
\(\displaystyle \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-4} \, dx\) |
\(\Big \downarrow \) 907 |
\(\displaystyle \frac {\left (\frac {b c}{b c n-a d n}+3\right ) \int \left (b x^n+a\right )^3 \left (d x^n+c\right )^{-4-\frac {1}{n}}dx}{3 a}-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{3 a n (b c-a d)}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {\left (\frac {b c}{b c n-a d n}+3\right ) \left (\frac {3 a n \int \left (b x^n+a\right )^2 \left (d x^n+c\right )^{-3-\frac {1}{n}}dx}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\right )}{3 a}-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{3 a n (b c-a d)}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {\left (\frac {b c}{b c n-a d n}+3\right ) \left (\frac {3 a n \left (\frac {2 a n \int \left (b x^n+a\right ) \left (d x^n+c\right )^{-2-\frac {1}{n}}dx}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\right )}{3 a}-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{3 a n (b c-a d)}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {\left (\frac {b c}{b c n-a d n}+3\right ) \left (\frac {3 a n \left (\frac {2 a n \left (\frac {a n \int \left (d x^n+c\right )^{-1-\frac {1}{n}}dx}{c (n+1)}+\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}\right )}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\right )}{3 a}-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{3 a n (b c-a d)}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {\left (\frac {b c}{b c n-a d n}+3\right ) \left (\frac {3 a n \left (\frac {2 a n \left (\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}\right )}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\right )}{3 a}-\frac {b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{3 a n (b c-a d)}\) |
-1/3*(b*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(a*(b*c - a*d)*n) + ((3 + (b*c)/(b*c*n - a*d*n))*((x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(c* (1 + 3*n)) + (3*a*n*((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))))/(c*(1 + 3*n))))/(3 *a)
3.4.30.3.1 Defintions of rubi rules used
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_)*((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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1058\) vs. \(2(319)=638\).
Time = 4.52 (sec) , antiderivative size = 1059, normalized size of antiderivative = 3.24
(24*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^3*d*n^3+3*x*(x^n)^2*(c+d*x^n)^(-(1+ 4*n)/n)*a^2*c^2*d^2*n+26*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^3*d*n^2+12*x*x ^n*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^4*n^2+2*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a *b*c^3*d+9*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^3*d*n+10*x*x^n*(c+d*x^n)^(-( 1+4*n)/n)*a*b*c^4*n+16*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^2*d^2*n^2+4* x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a*b*c*d^3*n^2+24*x*(x^n)^3*(c+d*x^n)^(-(1 +4*n)/n)*a^2*c*d^3*n^3+x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^2*d^2*n+6*x* (x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*c*d^3*n^2+4*x*(x^n)^3*(c+d*x^n)^(-(1+4* n)/n)*b^2*c^3*d*n^2+4*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^2*d^2*n+24*x* (x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^3*d*n^2+x*(x^n)^4*(c+d*x^n)^(-(1+4*n) /n)*b^2*c^2*d^2*n^2+14*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^3*d*n+36*x*( x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^2*d^2*n^3+5*x*(x^n)^3*(c+d*x^n)^(-(1+4 *n)/n)*b^2*c^3*d*n+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^2*d^2*n^2+6*x *(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a^2*d^4*n^3+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n )/n)*b^2*c^4*n^2+x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^3*d+4*x*(x^n)^2*(c +d*x^n)^(-(1+4*n)/n)*b^2*c^4*n+x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^3*d+2*x* x^n*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^4+x*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^4+6*x*(c +d*x^n)^(-(1+4*n)/n)*a^2*c^4*n^3+x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^4+ 11*x*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^4*n^2+6*x*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^4 *n)/(2*n^2+3*n+1)/(1+3*n)/c^4
Time = 0.27 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.22 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {{\left (6 \, a^{2} d^{4} n^{3} + b^{2} c^{2} d^{2} n + {\left (b^{2} c^{2} d^{2} + 4 \, a b c d^{3}\right )} n^{2}\right )} x x^{4 \, n} + {\left (24 \, a^{2} c d^{3} n^{3} + b^{2} c^{3} d + 2 \, {\left (2 \, b^{2} c^{3} d + 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} n^{2} + {\left (5 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + {\left (36 \, a^{2} c^{2} d^{2} n^{3} + b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, {\left (b^{2} c^{4} + 8 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} n^{2} + {\left (4 \, b^{2} c^{4} + 14 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{2} c^{3} d n^{3} + 2 \, a b c^{4} + a^{2} c^{3} d + 2 \, {\left (6 \, a b c^{4} + 13 \, a^{2} c^{3} d\right )} n^{2} + {\left (10 \, a b c^{4} + 9 \, a^{2} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{2} c^{4} n^{3} + 11 \, a^{2} c^{4} n^{2} + 6 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \]
((6*a^2*d^4*n^3 + b^2*c^2*d^2*n + (b^2*c^2*d^2 + 4*a*b*c*d^3)*n^2)*x*x^(4* n) + (24*a^2*c*d^3*n^3 + b^2*c^3*d + 2*(2*b^2*c^3*d + 8*a*b*c^2*d^2 + 3*a^ 2*c*d^3)*n^2 + (5*b^2*c^3*d + 4*a*b*c^2*d^2)*n)*x*x^(3*n) + (36*a^2*c^2*d^ 2*n^3 + b^2*c^4 + 2*a*b*c^3*d + 3*(b^2*c^4 + 8*a*b*c^3*d + 7*a^2*c^2*d^2)* n^2 + (4*b^2*c^4 + 14*a*b*c^3*d + 3*a^2*c^2*d^2)*n)*x*x^(2*n) + (24*a^2*c^ 3*d*n^3 + 2*a*b*c^4 + a^2*c^3*d + 2*(6*a*b*c^4 + 13*a^2*c^3*d)*n^2 + (10*a *b*c^4 + 9*a^2*c^3*d)*n)*x*x^n + (6*a^2*c^4*n^3 + 11*a^2*c^4*n^2 + 6*a^2*c ^4*n + a^2*c^4)*x)/((6*c^4*n^3 + 11*c^4*n^2 + 6*c^4*n + c^4)*(d*x^n + c)^( (4*n + 1)/n))
Leaf count of result is larger than twice the leaf count of optimal. 2746 vs. \(2 (282) = 564\).
Time = 13.32 (sec) , antiderivative size = 2746, normalized size of antiderivative = 8.40 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Too large to display} \]
6*a**2*c**3*c**(1/n)*c**(-4 - 1/n)*n**3*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/ (d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x* *n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x** n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 11*a**2*c**3*c**(1/n)*c**(-4 - 1/n)*n**2*gamma (1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2* d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d** (1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n) *n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**2*c**3*c**(1 /n)*c**(-4 - 1/n)*n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n) *gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamm a(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma (4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + a**2*c**3*c**(1/n)*c**(-4 - 1/n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c /(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x **n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x* *n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 18*a**2*c**2*c**(1/n)*c**(-4 - 1/n)*d*n**3*x* *n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3...
\[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 4} \,d x } \]
Exception generated. \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \]
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%%%{27,[1,0,4,3,1,3,2,0]%%%}+%%%{27,[1,0,4,2,1,3,2,0]%%%}+%%%{ 9,[1,0,4,
Timed out. \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+4}} \,d x \]