Integrand size = 26, antiderivative size = 105 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\frac {a}{2 b (b c-a d) n \left (a+b x^n\right )^2}-\frac {c}{(b c-a d)^2 n \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{(b c-a d)^3 n}+\frac {c d \log \left (c+d x^n\right )}{(b c-a d)^3 n} \] Output:
1/2*a/b/(-a*d+b*c)/n/(a+b*x^n)^2-c/(-a*d+b*c)^2/n/(a+b*x^n)-c*d*ln(a+b*x^n )/(-a*d+b*c)^3/n+c*d*ln(c+d*x^n)/(-a*d+b*c)^3/n
Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\frac {-a b c-a^2 d-2 b^2 c x^n}{2 b (b c-a d)^2 n \left (a+b x^n\right )^2}-\frac {c d \log \left (a+b x^n\right )}{(b c-a d)^3 n}+\frac {c d \log \left (c+d x^n\right )}{(b c-a d)^3 n} \] Input:
Integrate[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]
Output:
(-(a*b*c) - a^2*d - 2*b^2*c*x^n)/(2*b*(b*c - a*d)^2*n*(a + b*x^n)^2) - (c* d*Log[a + b*x^n])/((b*c - a*d)^3*n) + (c*d*Log[c + d*x^n])/((b*c - a*d)^3* n)
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {948, 86, 2009}
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.
\(\displaystyle \int \frac {x^{2 n-1}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int \frac {x^n}{\left (b x^n+a\right )^3 \left (d x^n+c\right )}dx^n}{n}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\int \left (\frac {c d^2}{(b c-a d)^3 \left (d x^n+c\right )}-\frac {b c d}{(b c-a d)^3 \left (b x^n+a\right )}+\frac {b c}{(b c-a d)^2 \left (b x^n+a\right )^2}-\frac {a}{(b c-a d) \left (b x^n+a\right )^3}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a}{2 b (b c-a d) \left (a+b x^n\right )^2}-\frac {c}{(b c-a d)^2 \left (a+b x^n\right )}-\frac {c d \log \left (a+b x^n\right )}{(b c-a d)^3}+\frac {c d \log \left (c+d x^n\right )}{(b c-a d)^3}}{n}\) |
Input:
Int[x^(-1 + 2*n)/((a + b*x^n)^3*(c + d*x^n)),x]
Output:
(a/(2*b*(b*c - a*d)*(a + b*x^n)^2) - c/((b*c - a*d)^2*(a + b*x^n)) - (c*d* Log[a + b*x^n])/(b*c - a*d)^3 + (c*d*Log[c + d*x^n])/(b*c - a*d)^3)/n
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 1.83 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.50
method | result | size |
risch | \(-\frac {2 b^{2} c \,x^{n}+a^{2} d +a b c}{2 \left (a d -c b \right )^{2} b n \left (a +b \,x^{n}\right )^{2}}-\frac {c d \ln \left (x^{n}+\frac {c}{d}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c d \ln \left (x^{n}+\frac {a}{b}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(157\) |
norman | \(\frac {-\frac {b c \,{\mathrm e}^{n \ln \left (x \right )}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) n}+\frac {a \left (-a b d -b^{2} c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} n}}{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}+\frac {c d \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c d \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(203\) |
Input:
int(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x,method=_RETURNVERBOSE)
Output:
-1/2*(2*b^2*c*x^n+a^2*d+a*b*c)/(a*d-b*c)^2/b/n/(a+b*x^n)^2-c*d/n/(a^3*d^3- 3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(x^n+c/d)+c*d/n/(a^3*d^3-3*a^2*b*c* d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(x^n+a/b)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (103) = 206\).
Time = 0.11 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.54 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=-\frac {a b^{2} c^{2} - a^{3} d^{2} + 2 \, {\left (b^{3} c^{2} - a b^{2} c d\right )} x^{n} + 2 \, {\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (b x^{n} + a\right ) - 2 \, {\left (b^{3} c d x^{2 \, n} + 2 \, a b^{2} c d x^{n} + a^{2} b c d\right )} \log \left (d x^{n} + c\right )}{2 \, {\left ({\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} n x^{2 \, n} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} n x^{n} + {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} n\right )}} \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x, algorithm="fricas")
Output:
-1/2*(a*b^2*c^2 - a^3*d^2 + 2*(b^3*c^2 - a*b^2*c*d)*x^n + 2*(b^3*c*d*x^(2* n) + 2*a*b^2*c*d*x^n + a^2*b*c*d)*log(b*x^n + a) - 2*(b^3*c*d*x^(2*n) + 2* a*b^2*c*d*x^n + a^2*b*c*d)*log(d*x^n + c))/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a ^2*b^4*c*d^2 - a^3*b^3*d^3)*n*x^(2*n) + 2*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3 *a^3*b^3*c*d^2 - a^4*b^2*d^3)*n*x^n + (a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a ^4*b^2*c*d^2 - a^5*b*d^3)*n)
Exception generated. \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(x**(-1+2*n)/(a+b*x**n)**3/(c+d*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).
Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.31 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=-\frac {c d \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} + \frac {c d \log \left (\frac {d x^{n} + c}{d}\right )}{b^{3} c^{3} n - 3 \, a b^{2} c^{2} d n + 3 \, a^{2} b c d^{2} n - a^{3} d^{3} n} - \frac {2 \, b^{2} c x^{n} + a b c + a^{2} d}{2 \, {\left (a^{2} b^{3} c^{2} n - 2 \, a^{3} b^{2} c d n + a^{4} b d^{2} n + {\left (b^{5} c^{2} n - 2 \, a b^{4} c d n + a^{2} b^{3} d^{2} n\right )} x^{2 \, n} + 2 \, {\left (a b^{4} c^{2} n - 2 \, a^{2} b^{3} c d n + a^{3} b^{2} d^{2} n\right )} x^{n}\right )}} \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x, algorithm="maxima")
Output:
-c*d*log((b*x^n + a)/b)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b*c*d^2*n - a ^3*d^3*n) + c*d*log((d*x^n + c)/d)/(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2*b* c*d^2*n - a^3*d^3*n) - 1/2*(2*b^2*c*x^n + a*b*c + a^2*d)/(a^2*b^3*c^2*n - 2*a^3*b^2*c*d*n + a^4*b*d^2*n + (b^5*c^2*n - 2*a*b^4*c*d*n + a^2*b^3*d^2*n )*x^(2*n) + 2*(a*b^4*c^2*n - 2*a^2*b^3*c*d*n + a^3*b^2*d^2*n)*x^n)
\[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\int { \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x, algorithm="giac")
Output:
integrate(x^(2*n - 1)/((b*x^n + a)^3*(d*x^n + c)), x)
Timed out. \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\int \frac {x^{2\,n-1}}{{\left (a+b\,x^n\right )}^3\,\left (c+d\,x^n\right )} \,d x \] Input:
int(x^(2*n - 1)/((a + b*x^n)^3*(c + d*x^n)),x)
Output:
int(x^(2*n - 1)/((a + b*x^n)^3*(c + d*x^n)), x)
Time = 0.22 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.10 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3 \left (c+d x^n\right )} \, dx=\frac {2 x^{2 n} \mathrm {log}\left (x^{n} b +a \right ) a \,b^{3} c d -2 x^{2 n} \mathrm {log}\left (x^{n} d +c \right ) a \,b^{3} c d +x^{2 n} a \,b^{3} c d -x^{2 n} b^{4} c^{2}+4 x^{n} \mathrm {log}\left (x^{n} b +a \right ) a^{2} b^{2} c d -4 x^{n} \mathrm {log}\left (x^{n} d +c \right ) a^{2} b^{2} c d +2 \,\mathrm {log}\left (x^{n} b +a \right ) a^{3} b c d -2 \,\mathrm {log}\left (x^{n} d +c \right ) a^{3} b c d -a^{4} d^{2}+a^{3} b c d}{2 a b n \left (x^{2 n} a^{3} b^{2} d^{3}-3 x^{2 n} a^{2} b^{3} c \,d^{2}+3 x^{2 n} a \,b^{4} c^{2} d -x^{2 n} b^{5} c^{3}+2 x^{n} a^{4} b \,d^{3}-6 x^{n} a^{3} b^{2} c \,d^{2}+6 x^{n} a^{2} b^{3} c^{2} d -2 x^{n} a \,b^{4} c^{3}+a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -a^{2} b^{3} c^{3}\right )} \] Input:
int(x^(-1+2*n)/(a+b*x^n)^3/(c+d*x^n),x)
Output:
(2*x**(2*n)*log(x**n*b + a)*a*b**3*c*d - 2*x**(2*n)*log(x**n*d + c)*a*b**3 *c*d + x**(2*n)*a*b**3*c*d - x**(2*n)*b**4*c**2 + 4*x**n*log(x**n*b + a)*a **2*b**2*c*d - 4*x**n*log(x**n*d + c)*a**2*b**2*c*d + 2*log(x**n*b + a)*a* *3*b*c*d - 2*log(x**n*d + c)*a**3*b*c*d - a**4*d**2 + a**3*b*c*d)/(2*a*b*n *(x**(2*n)*a**3*b**2*d**3 - 3*x**(2*n)*a**2*b**3*c*d**2 + 3*x**(2*n)*a*b** 4*c**2*d - x**(2*n)*b**5*c**3 + 2*x**n*a**4*b*d**3 - 6*x**n*a**3*b**2*c*d* *2 + 6*x**n*a**2*b**3*c**2*d - 2*x**n*a*b**4*c**3 + a**5*d**3 - 3*a**4*b*c *d**2 + 3*a**3*b**2*c**2*d - a**2*b**3*c**3))