\(\int \frac {(f x)^m (a+c x^{2 n})^p}{(d+e x^n)^3} \, dx\) [92]

Optimal result

Integrand size = 26, antiderivative size = 412 \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2 n},-p,3,1+\frac {1+m}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+m)}-\frac {3 e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+n}{2 n},-p,3,\frac {1+m+3 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+m+n)}+\frac {3 e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+2 n}{2 n},-p,3,\frac {1+m+4 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^5 (1+m+2 n)}-\frac {e^3 x^{1+3 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+3 n}{2 n},-p,3,\frac {1+m+5 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^6 (1+m+3 n)} \]

[Out]

x*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,3,-p,1+1/2*(1+m)/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^3/(1+m)/((1+
c*x^(2*n)/a)^p)-3*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,3,-p,1/2*(1+m+3*n)/n,e^2*x^(2*n)/d^
2,-c*x^(2*n)/a)/d^4/(1+m+n)/((1+c*x^(2*n)/a)^p)+3*e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+2*n)
/n,3,-p,1/2*(1+m+4*n)/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^5/(1+m+2*n)/((1+c*x^(2*n)/a)^p)-e^3*x^(1+3*n)*(f*x)^m*
(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+3*n)/n,3,-p,1/2*(1+m+5*n)/n,e^2*x^(2*n)/d^2,-c*x^(2*n)/a)/d^6/(1+m+3*n)/((1+
c*x^(2*n)/a)^p)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1576, 525, 524} \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=-\frac {e^3 x^{3 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+3 n+1}{2 n},-p,3,\frac {m+5 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^6 (m+3 n+1)}+\frac {3 e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2 n+1}{2 n},-p,3,\frac {m+4 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^5 (m+2 n+1)}-\frac {3 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+n+1}{2 n},-p,3,\frac {m+3 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (m+n+1)}+\frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2 n},-p,3,\frac {m+1}{2 n}+1,-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (m+1)} \]

[In]

Int[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^3,x]

[Out]

(x*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m)/(2*n), -p, 3, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))
/d^2])/(d^3*(1 + m)*(1 + (c*x^(2*n))/a)^p) - (3*e*x^(1 + n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + n)/(2*
n), -p, 3, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + m + n)*(1 + (c*x^(2*n))/a)^p)
+ (3*e^2*x^(1 + 2*n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + 2*n)/(2*n), -p, 3, (1 + m + 4*n)/(2*n), -((c*
x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^5*(1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p) - (e^3*x^(1 + 3*n)*(f*x)^m*(a + c*x
^(2*n))^p*AppellF1[(1 + m + 3*n)/(2*n), -p, 3, (1 + m + 5*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^6
*(1 + m + 3*n)*(1 + (c*x^(2*n))/a)^p)

Rule 524

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] /; Free
Q[{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])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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])

Rule 1576

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (f x)^m\right ) \int \left (\frac {d^3 x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^3}+\frac {3 d^2 e x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}-\frac {3 d e^2 x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}+\frac {e^3 x^{m+3 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3}\right ) \, dx \\ & = \left (d^3 x^{-m} (f x)^m\right ) \int \frac {x^m \left (a+c x^{2 n}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^3} \, dx+\left (3 d^2 e x^{-m} (f x)^m\right ) \int \frac {x^{m+n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx-\left (3 d e^2 x^{-m} (f x)^m\right ) \int \frac {x^{m+2 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx+\left (e^3 x^{-m} (f x)^m\right ) \int \frac {x^{m+3 n} \left (a+c x^{2 n}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx \\ & = \left (d^3 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^m \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (d^2-e^2 x^{2 n}\right )^3} \, dx+\left (3 d^2 e x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{m+n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx-\left (3 d e^2 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{m+2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx+\left (e^3 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^{m+3 n} \left (1+\frac {c x^{2 n}}{a}\right )^p}{\left (-d^2+e^2 x^{2 n}\right )^3} \, dx \\ & = \frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m}{2 n};-p,3;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+m)}-\frac {3 e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m+n}{2 n};-p,3;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+m+n)}+\frac {3 e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m+2 n}{2 n};-p,3;\frac {1+m+4 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^5 (1+m+2 n)}-\frac {e^3 x^{1+3 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+m+3 n}{2 n};-p,3;\frac {1+m+5 n}{2 n};-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^6 (1+m+3 n)} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx \]

[In]

Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^3,x]

[Out]

Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^3, x]

Maple [F]

\[\int \frac {\left (f x \right )^{m} \left (a +c \,x^{2 n}\right )^{p}}{\left (d +e \,x^{n}\right )^{3}}d x\]

[In]

int((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x)

[Out]

int((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x)

Fricas [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{3}} \,d x } \]

[In]

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x, algorithm="fricas")

[Out]

integral((c*x^(2*n) + a)^p*(f*x)^m/(e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x)**m*(a+c*x**(2*n))**p/(d+e*x**n)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{3}} \,d x } \]

[In]

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x, algorithm="maxima")

[Out]

integrate((c*x^(2*n) + a)^p*(f*x)^m/(e*x^n + d)^3, x)

Giac [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{3}} \,d x } \]

[In]

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^3,x, algorithm="giac")

[Out]

integrate((c*x^(2*n) + a)^p*(f*x)^m/(e*x^n + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^3} \, dx=\int \frac {{\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m}{{\left (d+e\,x^n\right )}^3} \,d x \]

[In]

int(((a + c*x^(2*n))^p*(f*x)^m)/(d + e*x^n)^3,x)

[Out]

int(((a + c*x^(2*n))^p*(f*x)^m)/(d + e*x^n)^3, x)