\(\int \frac {(a+b x^n)^p}{(c+d x^n)^3} \, dx\) [319]

Optimal result

Integrand size = 19, antiderivative size = 59 \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^3} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (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 = {441, 440} \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^3} \]

[In]

Int[(a + b*x^n)^p/(c + d*x^n)^3,x]

[Out]

(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)

Rule 440

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])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[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])

Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^n}{a}\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} F_1\left (\frac {1}{n};-p,3;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(59)=118\).

Time = 0.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.05 \[ \int \frac {\left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\frac {a c (1+n) x \left (a+b x^n\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\left (c+d x^n\right )^3 \left (b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,3,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-3 a d n x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,4,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,3,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \]

[In]

Integrate[(a + b*x^n)^p/(c + d*x^n)^3,x]

[Out]

(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)*AppellF1[n^(-1), -p, 3, 1 + n^(-1), -((b*
x^n)/a), -((d*x^n)/c)]))

Maple [F]

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

[In]

int((a+b*x^n)^p/(c+d*x^n)^3,x)

[Out]

int((a+b*x^n)^p/(c+d*x^n)^3,x)

Fricas [F]

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

[In]

integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="fricas")

[Out]

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)

Sympy [F]

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

[In]

integrate((a+b*x**n)**p/(c+d*x**n)**3,x)

[Out]

Integral((a + b*x**n)**p/(c + d*x**n)**3, x)

Maxima [F]

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

[In]

integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p/(d*x^n + c)^3, x)

Giac [F]

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

[In]

integrate((a+b*x^n)^p/(c+d*x^n)^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p/(d*x^n + c)^3, x)

Mupad [F(-1)]

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

[In]

int((a + b*x^n)^p/(c + d*x^n)^3,x)

[Out]

int((a + b*x^n)^p/(c + d*x^n)^3, x)