Integrand size = 25, antiderivative size = 178 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 (1+n) (1+2 n) (1+3 n)}+\frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (1+n) (1+2 n) (1+3 n)} \]
x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/c/(1+3*n)+3*a*n*x*(a+b*x^n)^2*(c+d*x^n)^( -2-1/n)/c^2/(6*n^2+5*n+1)+6*a^2*n^2*x*(a+b*x^n)*(c+d*x^n)^(-1-1/n)/c^3/(6* n^3+11*n^2+6*n+1)+6*a^3*n^3*x/c^4/(6*n^3+11*n^2+6*n+1)/((c+d*x^n)^(1/n))
Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.22 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-3-\frac {1}{n}} \left (b^3 c^3 \left (1+3 n+2 n^2\right ) x^{3 n}+3 a b^2 c^2 (1+n) x^{2 n} \left (c+3 c n+d n x^n\right )+3 a^2 b c x^n \left (c^2 \left (1+5 n+6 n^2\right )+2 c d n (1+3 n) x^n+2 d^2 n^2 x^{2 n}\right )+a^3 \left (c^3 \left (1+6 n+11 n^2+6 n^3\right )+3 c^2 d n \left (1+5 n+6 n^2\right ) x^n+6 c d^2 n^2 (1+3 n) x^{2 n}+6 d^3 n^3 x^{3 n}\right )\right )}{c^4 (1+n) (1+2 n) (1+3 n)} \]
(x*(c + d*x^n)^(-3 - n^(-1))*(b^3*c^3*(1 + 3*n + 2*n^2)*x^(3*n) + 3*a*b^2* c^2*(1 + n)*x^(2*n)*(c + 3*c*n + d*n*x^n) + 3*a^2*b*c*x^n*(c^2*(1 + 5*n + 6*n^2) + 2*c*d*n*(1 + 3*n)*x^n + 2*d^2*n^2*x^(2*n)) + a^3*(c^3*(1 + 6*n + 11*n^2 + 6*n^3) + 3*c^2*d*n*(1 + 5*n + 6*n^2)*x^n + 6*c*d^2*n^2*(1 + 3*n)* x^(2*n) + 6*d^3*n^3*x^(3*n))))/(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n))
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {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.
\(\displaystyle \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-4} \, dx\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {3 a n \int \left (b x^n+a\right )^2 \left (d x^n+c\right )^{-3-\frac {1}{n}}dx}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {3 a n \left (\frac {2 a n \int \left (b x^n+a\right ) \left (d x^n+c\right )^{-2-\frac {1}{n}}dx}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {3 a n \left (\frac {2 a n \left (\frac {a n \int \left (d x^n+c\right )^{-1-\frac {1}{n}}dx}{c (n+1)}+\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}\right )}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \frac {3 a n \left (\frac {2 a n \left (\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}\right )}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\) |
(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))
3.4.21.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1287\) vs. \(2(174)=348\).
Time = 5.02 (sec) , antiderivative size = 1288, normalized size of antiderivative = 7.24
(6*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a^3*d^4*n^3+2*x*(x^n)^3*(c+d*x^n)^(-(1 +4*n)/n)*b^3*c^4*n^2+x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^3*d+3*x*(x^n)^ 3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4*n+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b^ 2*c^4+x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d+3*x*x^n*(c+d*x^n)^(-(1+4*n)/n )*a^2*b*c^4+2*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^3*d*n^2+24*x*(x^n)^3* (c+d*x^n)^(-(1+4*n)/n)*a^3*c*d^3*n^3+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^ 3*c^3*d*n+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^3*c*d^3*n^2+36*x*(x^n)^2*(c +d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2*n^3+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a ^3*c^2*d^2*n^2+9*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^4*n^2+24*x*x^n*( c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d*n^3+3*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b ^2*c^3*d+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2*n+12*x*(x^n)^2*(c+ d*x^n)^(-(1+4*n)/n)*a*b^2*c^4*n+26*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d* n^2+18*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4*n^2+3*x*(x^n)^2*(c+d*x^n)^(- (1+4*n)/n)*a^2*b*c^3*d+9*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d*n+6*x*(x^n )^4*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c*d^3*n^2+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n) /n)*a*b^2*c^2*d^2*n^2+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^2*d^2*n+2 4*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^2*d^2*n^2+12*x*(x^n)^3*(c+d*x^n )^(-(1+4*n)/n)*a*b^2*c^3*d*n^2+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^ 2*d^2*n+15*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d*n+36*x*(x^n)^2*(c+ d*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d*n^2+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)...
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (174) = 348\).
Time = 0.27 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.69 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {{\left (6 \, a^{3} d^{4} n^{3} + b^{3} c^{3} d + {\left (2 \, b^{3} c^{3} d + 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2}\right )} n\right )} x x^{4 \, n} + {\left (24 \, a^{3} c d^{3} n^{3} + b^{3} c^{4} + 3 \, a b^{2} c^{3} d + 2 \, {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 12 \, a^{2} b c^{2} d^{2} + 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (12 \, a^{3} c^{2} d^{2} n^{3} + a b^{2} c^{4} + a^{2} b c^{3} d + {\left (3 \, a b^{2} c^{4} + 12 \, a^{2} b c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} n^{2} + {\left (4 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{3} c^{3} d n^{3} + 3 \, a^{2} b c^{4} + a^{3} c^{3} d + 2 \, {\left (9 \, a^{2} b c^{4} + 13 \, a^{3} c^{3} d\right )} n^{2} + 3 \, {\left (5 \, a^{2} b c^{4} + 3 \, a^{3} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{3} c^{4} n^{3} + 11 \, a^{3} c^{4} n^{2} + 6 \, a^{3} c^{4} n + a^{3} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \]
((6*a^3*d^4*n^3 + b^3*c^3*d + (2*b^3*c^3*d + 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d ^3)*n^2 + 3*(b^3*c^3*d + a*b^2*c^2*d^2)*n)*x*x^(4*n) + (24*a^3*c*d^3*n^3 + b^3*c^4 + 3*a*b^2*c^3*d + 2*(b^3*c^4 + 6*a*b^2*c^3*d + 12*a^2*b*c^2*d^2 + 3*a^3*c*d^3)*n^2 + 3*(b^3*c^4 + 5*a*b^2*c^3*d + 2*a^2*b*c^2*d^2)*n)*x*x^( 3*n) + 3*(12*a^3*c^2*d^2*n^3 + a*b^2*c^4 + a^2*b*c^3*d + (3*a*b^2*c^4 + 12 *a^2*b*c^3*d + 7*a^3*c^2*d^2)*n^2 + (4*a*b^2*c^4 + 7*a^2*b*c^3*d + a^3*c^2 *d^2)*n)*x*x^(2*n) + (24*a^3*c^3*d*n^3 + 3*a^2*b*c^4 + a^3*c^3*d + 2*(9*a^ 2*b*c^4 + 13*a^3*c^3*d)*n^2 + 3*(5*a^2*b*c^4 + 3*a^3*c^3*d)*n)*x*x^n + (6* a^3*c^4*n^3 + 11*a^3*c^4*n^2 + 6*a^3*c^4*n + a^3*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. 2822 vs. \(2 (160) = 320\).
Time = 29.87 (sec) , antiderivative size = 2822, normalized size of antiderivative = 15.85 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Too large to display} \]
6*a**3*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**3*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**3*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**3*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**3*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...
\[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 4} \,d x } \]
Exception generated. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \]
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%%%{81,[2,0,6,4,2,4,3,0]%%%}+%%%{108,[2,0,6,3,2,4,3,0]%%%}+%%% {54,[2,0,
Timed out. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^3}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+4}} \,d x \]