Integrand size = 19, antiderivative size = 59 \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^3} \] Output:
x*(a+b*x^n)^p*AppellF1(1/n,-p,3,1+1/n,-b*x^n/a,-d*x^n/c)/c^3/((1+b*x^n/a)^ p)
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(59)=118\).
Time = 0.50 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {a c (1+n) x \left (a+b x^n\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\left (c+d x^n\right )^3 \left (b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,3,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,4,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \] Input:
Integrate[(a + b*x^n)^p/(c + d*x^n)^3,x]
Output:
(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 3, 1 + n^(-1), -((b*x^n) /a), -((d*x^n)/c)])/((c + d*x^n)^3*(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p , 3, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] - 3*a*d*n*x^n*AppellF1[1 + n^ (-1), -p, 4, 2 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF 1[n^(-1), -p, 3, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]))
Time = 0.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {937, 936}
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.
\(\displaystyle \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {\left (\frac {b x^n}{a}+1\right )^p}{\left (d x^n+c\right )^3}dx\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^3}\) |
Input:
Int[(a + b*x^n)^p/(c + d*x^n)^3,x]
Output:
(x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 3, 1 + n^(-1), -((b*x^n)/a), -((d*x^ n)/c)])/(c^3*(1 + (b*x^n)/a)^p)
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((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[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {\left (a +b \,x^{n}\right )^{p}}{\left (c +d \,x^{n}\right )^{3}}d x\]
Input:
int((a+b*x^n)^p/(c+d*x^n)^3,x)
Output:
int((a+b*x^n)^p/(c+d*x^n)^3,x)
\[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="fricas")
Output:
integral((b*x^n + a)^p/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3) , x)
\[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int \frac {\left (a + b x^{n}\right )^{p}}{\left (c + d x^{n}\right )^{3}}\, dx \] Input:
integrate((a+b*x**n)**p/(c+d*x**n)**3,x)
Output:
Integral((a + b*x**n)**p/(c + d*x**n)**3, x)
\[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="maxima")
Output:
integrate((b*x^n + a)^p/(d*x^n + c)^3, x)
\[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:
integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="giac")
Output:
integrate((b*x^n + a)^p/(d*x^n + c)^3, x)
Timed out. \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^3} \,d x \] Input:
int((a + b*x^n)^p/(c + d*x^n)^3,x)
Output:
int((a + b*x^n)^p/(c + d*x^n)^3, x)
\[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\int \frac {\left (x^{n} b +a \right )^{p}}{x^{3 n} d^{3}+3 x^{2 n} c \,d^{2}+3 x^{n} c^{2} d +c^{3}}d x \] Input:
int((a+b*x^n)^p/(c+d*x^n)^3,x)
Output:
int((x**n*b + a)**p/(x**(3*n)*d**3 + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d + c **3),x)