Integrand size = 29, antiderivative size = 210 \[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (-\frac {2}{n},1,-q,-\frac {2-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x^2}+\frac {c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} \operatorname {AppellF1}\left (-\frac {2}{n},1,-q,-\frac {2-n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x^2} \]
c*(d+e*x^n)^q*AppellF1(-2/n,1,-q,(-2+n)/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)), -e*x^n/d)/x^2/((1+e*x^n/d)^q)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))+c*(d+e*x^n) ^q*AppellF1(-2/n,1,-q,(-2+n)/n,-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)),-e*x^n/d)/x ^2/((1+e*x^n/d)^q)/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))
\[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx \]
Time = 0.43 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1880, 1013, 1012}
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.
\(\displaystyle \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx\) |
\(\Big \downarrow \) 1880 |
\(\displaystyle \frac {2 c \int \frac {\left (e x^n+d\right )^q}{x^3 \left (2 c x^n+b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {\left (e x^n+d\right )^q}{x^3 \left (2 c x^n+b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {2 c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \int \frac {\left (\frac {e x^n}{d}+1\right )^q}{x^3 \left (2 c x^n+b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \int \frac {\left (\frac {e x^n}{d}+1\right )^q}{x^3 \left (2 c x^n+b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {2}{n},1,-q,-\frac {2-n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {2}{n},1,-q,-\frac {2-n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}\) |
-((c*(d + e*x^n)^q*AppellF1[-2/n, 1, -q, -((2 - n)/n), (-2*c*x^n)/(b - Sqr t[b^2 - 4*a*c]), -((e*x^n)/d)])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c]) *x^2*(1 + (e*x^n)/d)^q)) + (c*(d + e*x^n)^q*AppellF1[-2/n, 1, -q, -((2 - n )/n), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^n)/d)])/(Sqrt[b^2 - 4*a*c ]*(b + Sqrt[b^2 - 4*a*c])*x^2*(1 + (e*x^n)/d)^q)
3.2.51.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^( n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[ 2*(c/r) Int[(f*x)^m*((d + e*x^n)^q/(b - r + 2*c*x^n)), x], x] - Simp[2*(c /r) Int[(f*x)^m*((d + e*x^n)^q/(b + r + 2*c*x^n)), x], x]] /; FreeQ[{a, b , c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && !Rati onalQ[n]
\[\int \frac {\left (d +e \,x^{n}\right )^{q}}{x^{3} \left (a +b \,x^{n}+c \,x^{2 n}\right )}d x\]
\[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int \frac {{\left (d+e\,x^n\right )}^q}{x^3\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]