Integrand size = 32, antiderivative size = 95 \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=-\frac {(a d-b c (2+p)) x^{-j (1+p)} \left (a x^j+b x^{j+n}\right )^{1+p}}{b^2 n (1+p) (2+p)}+\frac {d x^{n-j (1+p)} \left (a x^j+b x^{j+n}\right )^{1+p}}{b n (2+p)} \] Output:
-(a*d-b*c*(2+p))*(a*x^j+b*x^(j+n))^(p+1)/b^2/n/(p+1)/(2+p)/(x^(j*(p+1)))+d *x^(n-j*(p+1))*(a*x^j+b*x^(j+n))^(p+1)/b/n/(2+p)
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\frac {x^{-j p} \left (a+b x^n\right ) \left (x^j \left (a+b x^n\right )\right )^p \left (-a d+b c (2+p)+b d (1+p) x^n\right )}{b^2 n (1+p) (2+p)} \] Input:
Integrate[x^(-1 + n - j*p)*(c + d*x^n)*(a*x^j + b*x^(j + n))^p,x]
Output:
((a + b*x^n)*(x^j*(a + b*x^n))^p*(-(a*d) + b*c*(2 + p) + b*d*(1 + p)*x^n)) /(b^2*n*(1 + p)*(2 + p)*x^(j*p))
Time = 0.46 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1945, 1920}
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.
\(\displaystyle \int \left (c+d x^n\right ) x^{-j p+n-1} \left (a x^j+b x^{j+n}\right )^p \, dx\) |
\(\Big \downarrow \) 1945 |
\(\displaystyle \left (c-\frac {a d}{b (p+2)}\right ) \int x^{n-j p-1} \left (a x^j+b x^{j+n}\right )^pdx+\frac {d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {x^{-j (p+1)} \left (c-\frac {a d}{b (p+2)}\right ) \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+1)}+\frac {d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}\) |
Input:
Int[x^(-1 + n - j*p)*(c + d*x^n)*(a*x^j + b*x^(j + n))^p,x]
Output:
((c - (a*d)/(b*(2 + p)))*(a*x^j + b*x^(j + n))^(1 + p))/(b*n*(1 + p)*x^(j* (1 + p))) + (d*x^(n - j*(1 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(b*n*(2 + p))
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
\[\int x^{-j p +n -1} \left (c +d \,x^{n}\right ) \left (a \,x^{j}+b \,x^{j +n}\right )^{p}d x\]
Input:
int(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x)
Output:
int(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x)
Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.47 \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\frac {{\left ({\left (b^{2} d p + b^{2} d\right )} x x^{-j p + n - 1} x^{2 \, n} + {\left (2 \, b^{2} c + {\left (b^{2} c + a b d\right )} p\right )} x x^{-j p + n - 1} x^{n} + {\left (a b c p + 2 \, a b c - a^{2} d\right )} x x^{-j p + n - 1}\right )} \left (\frac {{\left (b x^{n} + a\right )} x^{j + n}}{x^{n}}\right )^{p}}{{\left (b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n\right )} x^{n}} \] Input:
integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="fricas" )
Output:
((b^2*d*p + b^2*d)*x*x^(-j*p + n - 1)*x^(2*n) + (2*b^2*c + (b^2*c + a*b*d) *p)*x*x^(-j*p + n - 1)*x^n + (a*b*c*p + 2*a*b*c - a^2*d)*x*x^(-j*p + n - 1 ))*((b*x^n + a)*x^(j + n)/x^n)^p/((b^2*n*p^2 + 3*b^2*n*p + 2*b^2*n)*x^n)
Timed out. \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\text {Timed out} \] Input:
integrate(x**(-j*p+n-1)*(c+d*x**n)*(a*x**j+b*x**(j+n))**p,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18 \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\frac {{\left (b x^{n} + a\right )} c e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{b n {\left (p + 1\right )}} + \frac {{\left (b^{2} {\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} d e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \] Input:
integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="maxima" )
Output:
(b*x^n + a)*c*e^(-j*p*log(x) + p*log(b*x^n + a) + p*log(x^j))/(b*n*(p + 1) ) + (b^2*(p + 1)*x^(2*n) + a*b*p*x^n - a^2)*d*e^(-j*p*log(x) + p*log(b*x^n + a) + p*log(x^j))/((p^2 + 3*p + 2)*b^2*n)
\[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\int { {\left (d x^{n} + c\right )} {\left (b x^{j + n} + a x^{j}\right )}^{p} x^{-j p + n - 1} \,d x } \] Input:
integrate(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x, algorithm="giac")
Output:
integrate((d*x^n + c)*(b*x^(j + n) + a*x^j)^p*x^(-j*p + n - 1), x)
Timed out. \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\int x^{n-j\,p-1}\,{\left (a\,x^j+b\,x^{j+n}\right )}^p\,\left (c+d\,x^n\right ) \,d x \] Input:
int(x^(n - j*p - 1)*(a*x^j + b*x^(j + n))^p*(c + d*x^n),x)
Output:
int(x^(n - j*p - 1)*(a*x^j + b*x^(j + n))^p*(c + d*x^n), x)
Time = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx=\frac {\left (x^{j +n} b +x^{j} a \right )^{p} \left (x^{2 n} b^{2} d p +x^{2 n} b^{2} d +x^{n} a b d p +x^{n} b^{2} c p +2 x^{n} b^{2} c -a^{2} d +a b c p +2 a b c \right )}{x^{j p} b^{2} n \left (p^{2}+3 p +2\right )} \] Input:
int(x^(-j*p+n-1)*(c+d*x^n)*(a*x^j+b*x^(j+n))^p,x)
Output:
((x**(j + n)*b + x**j*a)**p*(x**(2*n)*b**2*d*p + x**(2*n)*b**2*d + x**n*a* b*d*p + x**n*b**2*c*p + 2*x**n*b**2*c - a**2*d + a*b*c*p + 2*a*b*c))/(x**( j*p)*b**2*n*(p**2 + 3*p + 2))